The Focus on The Partitions Obtained By Parallel Moves In The Cancer's Extreme Recognition With Different Types of Extreme SuperHyperMatching Set and Polynomial on (Neutrosophic) SuperHyperGraphs

\documentclass[10pt,letterpaper]{article}
\usepackage[top=0.85in,left=2.79in,footskip=0.79in,marginparwidth=2in]{geometry}
\usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, tikz, color}
\usepackage[bookmarksnumbered, colorlinks, plainpages]{hyperref}
% use Unicode characters - try changing the option if you run into troubles with special characters (e.g. umlauts)
\usepackage[utf8]{inputenc}

% clean citations
\usepackage{cite}

% hyperref makes references clicky. use \url{www.example.com} or \href{www.example.com}{description} to add a clicky url
\usepackage{nameref,hyperref}

% line numbers
\usepackage[right]{lineno}

% improves typesetting in LaTeX
\usepackage{microtype}
\DisableLigatures[f]{encoding = *, family = * }

% text layout - change as needed
\raggedright
\setlength{\parindent}{0.5cm}
\textwidth 5.25in
\textheight 8.79in

% Remove % for double line spacing
%\usepackage{setspace}
%\doublespacing

% use adjustwidth environment to exceed text width (see examples in text)
\usepackage{changepage}

% adjust caption style
\usepackage[aboveskip=1pt,labelfont=bf,labelsep=period,singlelinecheck=off]{caption}

% remove brackets from references
\makeatletter
\renewcommand{\@biblabel}[1]{\quad#1.}
\makeatother

% headrule, footrule and page numbers
\usepackage{lastpage,fancyhdr,graphicx}
\usepackage{epstopdf}
\pagestyle{myheadings}
\pagestyle{fancy}
\fancyhf{}
\rfoot{\thepage/\pageref{LastPage}}
\renewcommand{\footrule}{\hrule height 2pt \vspace{2mm}}
\fancyheadoffset[L]{2.25in}
\fancyfootoffset[L]{2.25in}

% use \textcolor{color}{text} for colored text (e.g. highlight to-do areas)
\usepackage{color}

% define custom colors (this one is for figure captions)
\definecolor{Gray}{gray}{.25}

% this is required to include graphics
\usepackage{graphicx}

% use if you want to put caption to the side of the figure - see example in text
\usepackage{sidecap}
\usepackage{leftidx}
% use for have text wrap around figures
\usepackage{wrapfig}
\usepackage[pscoord]{eso-pic}
\usepackage[fulladjust]{marginnote}
\reversemarginpar
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\theoremstyle{observation}
\newtheorem{observation}[theorem]{Observation}
\theoremstyle{question}
\newtheorem{question}[theorem]{Question}
\theoremstyle{problem}
\newtheorem{problem}[theorem]{Problem}
\numberwithin{equation}{section}
% document begins here
\begin{document}
\vspace*{0.35in}
\linenumbers
% title goes here:
\begin{flushleft}
{\Large
\textbf\newline{
The Focus on The Partitions Obtained By Parallel Moves In The Cancer's Extreme Recognition With Different Types of Extreme SuperHyperMatching Set and Polynomial on (Neutrosophic) SuperHyperGraphs  
}
}
\newline
% authors go here:

Henry Garrett
\\
\bigskip
DrHenryGarrett@gmail.com
\\
\bigskip
Twitter's ID: @DrHenryGarrett $|$ \copyright DrHenryGarrett.wordpress.com

\end{flushleft}
\section{abstract}
In this research, new setting is introduced for new SuperHyperNotions, namely, a SuperHyperMatching  and Neutrosophic SuperHyperMatching . Two different types of SuperHyperDefinitions are debut for them but the research goes further and the SuperHyperNotion, SuperHyperUniform, and  SuperHyperClass based on that are well-defined and well-reviewed. The literature review is implemented in the whole of this research. For shining the elegancy and the significancy of this research, the comparison between this SuperHyperNotion with other SuperHyperNotions and fundamental SuperHyperNumbers are featured. The definitions are followed by the examples and the instances thus the clarifications are driven with different tools. The applications are figured out to make sense about the theoretical aspect of this ongoing research. The ``Cancer's Recognition'' are the under research to figure out the challenges make sense about ongoing and upcoming research. The special case is up. The cells are viewed in the deemed ways. There are different types of them. Some of them are individuals and some of them are well-modeled by the group of cells. These types are all officially called ``SuperHyperVertex'' but the relations amid them all officially called ``SuperHyperEdge''. The frameworks ``SuperHyperGraph'' and ``neutrosophic SuperHyperGraph'' are chosen and elected to research about ``Cancer's Recognition''. Thus these complex and dense SuperHyperModels open up some avenues to research on theoretical segments and ``Cancer's Recognition''. Some avenues are posed to pursue this research. It's also officially collected in the form of some questions and some problems. Assume a SuperHyperGraph.  Then an extreme SuperHyperMatching $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum cardinality of a SuperHyperSet $S$ of high cardinality SuperHyperEdges such that there's no SuperHyperVertex not to in a SuperHyperEdge and there's no SuperHyperEdge to have a SuperHyperVertex in a SuperHyperEdge; a neutrosophic SuperHyperMatching $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum neutrosophic cardinality of a neutrosophic SuperHyperSet $S$ of high neutrosophic cardinality neutrosophic  SuperHyperEdges such that there's no neutrosophic SuperHyperVertex not to in a neutrosophic SuperHyperEdge and there's no neutrosophic SuperHyperEdge to have a neutrosophic SuperHyperVertex in a neutrosophic SuperHyperEdge; an extreme SuperHyperMatching SuperHyperPolynomial $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the extreme SuperHyperPolynomial contains the coefficients defined as the number of the maximum cardinality of a SuperHyperSet $S$ of high cardinality SuperHyperEdges such that there's no SuperHyperVertex not to in a SuperHyperEdge and there's no SuperHyperEdge to have a SuperHyperVertex in a SuperHyperEdge and the power is corresponded to its coefficient; a neutrosophic SuperHyperMatching SuperHyperPolynomial $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the neutrosophic SuperHyperPolynomial contains the neutrosophic coefficients defined as the neutrosophic number of the maximum neutrosophic cardinality of a neutrosophic SuperHyperSet $S$ of high neutrosophic cardinality neutrosophic SuperHyperEdges such that there's no neutrosophic SuperHyperVertex not to in a neutrosophic SuperHyperEdge and there's no neutrosophic SuperHyperEdge to have a neutrosophic SuperHyperVertex in a neutrosophic SuperHyperEdge and the neutrosophic power is neutrosophicly corresponded to its neutrosophic coefficient; an extreme R-SuperHyperMatching $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum cardinality of a SuperHyperSet $S$ of high cardinality SuperHyperVertices such that there's no SuperHyperVertex not to in a SuperHyperEdge and there's no SuperHyperEdge to have a SuperHyperVertex in a SuperHyperEdge; a neutrosophic R-SuperHyperMatching $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum neutrosophic cardinality of a neutrosophic SuperHyperSet $S$ of high neutrosophic cardinality neutrosophic  SuperHyperVertices such that there's no neutrosophic SuperHyperVertex not to in a neutrosophic SuperHyperEdge and there's no neutrosophic SuperHyperEdge to have a neutrosophic SuperHyperVertex in a neutrosophic SuperHyperEdge; an extreme R-SuperHyperMatching SuperHyperPolynomial $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the extreme SuperHyperPolynomial contains the coefficients defined as the number of the maximum cardinality of a SuperHyperSet $S$ of high cardinality SuperHyperVertices such that there's no SuperHyperVertex not to in a SuperHyperEdge and there's no SuperHyperEdge to have a SuperHyperVertex in a SuperHyperEdge and the power is corresponded to its coefficient; a neutrosophic R-SuperHyperMatching SuperHyperPolynomial $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the neutrosophic SuperHyperPolynomial contains the neutrosophic coefficients defined as the neutrosophic number of the maximum neutrosophic cardinality of a neutrosophic SuperHyperSet $S$ of high neutrosophic cardinality  neutrosophic SuperHyperVertices such that there's no neutrosophic SuperHyperVertex not to in a neutrosophic SuperHyperEdge and there's no neutrosophic SuperHyperEdge to have a neutrosophic SuperHyperVertex in a neutrosophic SuperHyperEdge and the neutrosophic power is neutrosophicly corresponded to its neutrosophic coefficient.
Assume a SuperHyperGraph. Then $\delta-$SuperHyperMatching is a maximal    of SuperHyperVertices with a maximum cardinality such that either of the following expressions hold for the (neutrosophic) cardinalities of  SuperHyperNeighbors of $s\in S:$ there are $|S\cap N(s)| > |S\cap (V\setminus N(s))|+\delta;$ and $ |S\cap N(s)| < |S\cap (V\setminus N(s))|+\delta.$
The first Expression, holds if $S$ is an $\delta-$SuperHyperOffensive. And the second Expression, holds if $S$ is an $\delta-$SuperHyperDefensive; a neutrosophic $\delta-$SuperHyperMatching is a maximal neutrosophic     of SuperHyperVertices with maximum neutrosophic cardinality such that either of the following expressions hold for the neutrosophic cardinalities of  SuperHyperNeighbors of $s\in S$ there are: $|S\cap N(s)|_{neutrosophic} > |S\cap (V\setminus N(s))|_{neutrosophic}+\delta;$ and
$ |S\cap N(s)|_{neutrosophic} < |S\cap (V\setminus N(s))|_{neutrosophic}+\delta.$
The first Expression, holds if $S$ is a neutrosophic $\delta-$SuperHyperOffensive. And the second Expression, holds if $S$ is a neutrosophic $\delta-$SuperHyperDefensive
  It's useful to define a ``neutrosophic'' version of a SuperHyperMatching . Since there's more ways to get type-results to make a SuperHyperMatching  more understandable. For the sake of having neutrosophic SuperHyperMatching, there's a need to ``redefine'' the notion of a ``SuperHyperMatching ''. The SuperHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this procedure, there's the usage of the position of labels to assign to the values. Assume a SuperHyperMatching . It's redefined a neutrosophic SuperHyperMatching  if the mentioned Table holds, concerning, ``The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph'' with the key points,  
``The Values of The Vertices \& The Number of Position in Alphabet'',
``The Values of The SuperVertices\&The maximum Values of Its Vertices'',
``The Values of The Edges\&The maximum Values of Its Vertices'',
``The Values of The HyperEdges\&The maximum Values of Its Vertices'', ``The Values of The SuperHyperEdges\&The maximum Values of Its Endpoints''. To get structural examples and instances, I'm going to introduce the next SuperHyperClass of SuperHyperGraph based on a SuperHyperMatching . It's the main. It'll be disciplinary to have the foundation of previous definition in the kind of SuperHyperClass. If there's a need to have all SuperHyperConnectivities until the SuperHyperMatching, then it's officially called a ``SuperHyperMatching'' but otherwise, it isn't a SuperHyperMatching . There are some instances about the clarifications for the main definition titled a ``SuperHyperMatching ''. These two examples get more scrutiny and discernment since there are characterized in the disciplinary ways of the SuperHyperClass based on a SuperHyperMatching . For the sake of having a neutrosophic SuperHyperMatching, there's a need to ``redefine'' the notion of a ``neutrosophic SuperHyperMatching'' and a ``neutrosophic SuperHyperMatching ''. The SuperHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this procedure, there's the usage of the position of labels to assign to the values. Assume a neutrosophic SuperHyperGraph. It's redefined ``neutrosophic SuperHyperGraph'' if the intended Table holds. And a SuperHyperMatching  are redefined to a ``neutrosophic SuperHyperMatching'' if the intended Table holds. It's useful to define ``neutrosophic'' version of SuperHyperClasses. Since there's more ways to get neutrosophic type-results to make a neutrosophic SuperHyperMatching  more understandable. Assume a neutrosophic SuperHyperGraph. There are some neutrosophic SuperHyperClasses if the intended Table holds. Thus  SuperHyperPath,  SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultiPartite, and   SuperHyperWheel, are
``neutrosophic SuperHyperPath'', ``neutrosophic SuperHyperCycle'', ``neutrosophic SuperHyperStar'', ``neutrosophic SuperHyperBipartite'', ``neutrosophic SuperHyperMultiPartite'', and ``neutrosophic SuperHyperWheel'' if the intended Table holds.  A SuperHyperGraph has a ``neutrosophic SuperHyperMatching'' where it's the strongest [the maximum neutrosophic value from all the SuperHyperMatching  amid the maximum value amid all SuperHyperVertices from a SuperHyperMatching .] SuperHyperMatching . A graph is a SuperHyperUniform if it's a SuperHyperGraph and the number of elements of SuperHyperEdges are the same. Assume a neutrosophic SuperHyperGraph. There are some SuperHyperClasses as follows. It's SuperHyperPath if it's only one SuperVertex as intersection amid two given SuperHyperEdges with two exceptions; it's SuperHyperCycle if it's only one SuperVertex as intersection amid two given SuperHyperEdges; it's SuperHyperStar it's only one SuperVertex as intersection amid all SuperHyperEdges;
it's SuperHyperBipartite it's only one SuperVertex as intersection amid two given SuperHyperEdges and these SuperVertices, forming two separate sets, has no SuperHyperEdge in common; it's SuperHyperMultiPartite  it's only one SuperVertex as intersection amid two given SuperHyperEdges and these SuperVertices, forming multi separate sets, has no SuperHyperEdge in common; it's a SuperHyperWheel if it's only one SuperVertex as intersection amid two given SuperHyperEdges and one SuperVertex has one SuperHyperEdge with any common SuperVertex. The SuperHyperModel proposes the specific designs and the specific architectures. The SuperHyperModel is officially called ``SuperHyperGraph'' and ``Neutrosophic SuperHyperGraph''. In this SuperHyperModel, The ``specific'' cells and ``specific group'' of cells are SuperHyperModeled as ``SuperHyperVertices'' and the common and intended properties between ``specific'' cells and ``specific group'' of cells are SuperHyperModeled as ``SuperHyperEdges''. Sometimes, it's useful to have some degrees of determinacy, indeterminacy, and neutrality to have more precise SuperHyperModel which in this case the SuperHyperModel is called ``neutrosophic''. In the future research, the foundation will be based on the ``Cancer's Recognition'' and the results and the definitions will be introduced in redeemed ways. The recognition of the cancer in the long-term function. The specific region has been assigned by the model [it's called SuperHyperGraph] and the long cycle of the move from the cancer is identified by this research. Sometimes the move of the cancer hasn't be easily identified since there are some determinacy, indeterminacy and neutrality about the moves and the effects of the cancer on that region; this event leads us to choose another model [it's said to be neutrosophic SuperHyperGraph] to have convenient perception on what's happened and what's done. There are some specific models, which are well-known and they've got the names, and some SuperHyperGeneral SuperHyperModels. The moves and the traces of the cancer on the complex tracks and between complicated groups of cells could be fantasized by a neutrosophic SuperHyperPath(-/SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel). The aim is to find either the longest SuperHyperMatching  or the strongest SuperHyperMatching  in those neutrosophic SuperHyperModels. For the longest SuperHyperMatching, called SuperHyperMatching, and the strongest SuperHyperMatching, called neutrosophic SuperHyperMatching, some general results are introduced. Beyond that in SuperHyperStar, all possible SuperHyperPaths have only two SuperHyperEdges but it's not enough since it's essential to have at least three SuperHyperEdges to form any style of a SuperHyperCycle. There isn't any formation of any SuperHyperCycle but literarily, it's the deformation of any SuperHyperCycle. It, literarily, deforms and it doesn't form.  A basic familiarity with Extreme  SuperHyperMatching theory, SuperHyperGraphs, and Neutrosophic SuperHyperGraphs theory are proposed.
\\ \vspace{4mm}
\textbf{Keywords:} SuperHyperGraph, (Neutrosophic) SuperHyperMatching, Cancer's Recognition
   \\
\textbf{AMS Subject Classification:} 05C17, 05C22, 05E45
\section{Background}
There are some researches covering the topic of this research. In what follows, there are some discussion and literature reviews about them.
\\
First article is titled ``properties of SuperHyperGraph and neutrosophic SuperHyperGraph''  in \textbf{Ref.} \cite{HG1} by Henry Garrett (2022). It's first step toward the research on neutrosophic SuperHyperGraphs. This research article is published on the journal ``Neutrosophic Sets and Systems'' in issue 49 and the pages 531-561. In this research article, different types of notions like dominating, resolving, coloring, Eulerian(Hamiltonian) neutrosophic path, n-Eulerian(Hamiltonian) neutrosophic path, zero forcing number, zero forcing neutrosophic- number, independent number, independent neutrosophic-number, clique number, clique neutrosophic-number, matching number, matching neutrosophic-number, girth, neutrosophic girth, 1-zero-forcing number, 1-zero- forcing neutrosophic-number, failed 1-zero-forcing number, failed 1-zero-forcing neutrosophic-number, global- offensive alliance, t-offensive alliance, t-defensive alliance, t-powerful alliance, and global-powerful alliance are defined in SuperHyperGraph and neutrosophic SuperHyperGraph. Some Classes of SuperHyperGraph and Neutrosophic SuperHyperGraph are cases of research. Some results are applied in family of SuperHyperGraph and neutrosophic SuperHyperGraph. Thus this research article has concentrated on the vast notions and introducing the majority of notions.
\\
The seminal paper and groundbreaking article is titled
``neutrosophic co-degree and neutrosophic degree alongside chromatic numbers in the setting of some classes related to neutrosophic hypergraphs'' in \textbf{Ref.} \cite{HG2} by Henry Garrett (2022). In this research article, a novel approach is implemented on SuperHyperGraph and neutrosophic SuperHyperGraph based on general forms without using neutrosophic classes of neutrosophic SuperHyperGraph. It's published in prestigious and fancy journal is entitled “Journal of Current Trends in Computer Science Research (JCTCSR)” with abbreviation ``J Curr Trends Comp Sci Res'' in volume 1 and issue 1 with pages 06-14. The research article studies deeply with choosing neutrosophic hypergraphs instead of neutrosophic SuperHyperGraph. It's the breakthrough toward independent results based on initial background.
\\
The seminal paper and groundbreaking article is titled
``Super Hyper Dominating and Super Hyper Resolving on Neutrosophic Super Hyper Graphs and Their Directions in Game Theory and Neutrosophic Super Hyper Classes'' in \textbf{Ref.} \cite{HG3} by Henry Garrett (2022). In this research article, a novel approach is implemented on SuperHyperGraph and neutrosophic SuperHyperGraph based on fundamental SuperHyperNumber and using neutrosophic SuperHyperClasses of neutrosophic SuperHyperGraph. It's published in prestigious and fancy journal is entitled “Journal of Mathematical Techniques and Computational Mathematics(JMTCM)” with abbreviation ``J Math Techniques Comput Math'' in volume 1 and issue 3 with pages 242-263. The research article studies deeply with choosing directly neutrosophic SuperHyperGraph and SuperHyperGraph. It's the breakthrough toward independent results based on initial background and fundamental SuperHyperNumbers.
\\
In some articles are titled ``0039 | Closing Numbers and Super-Closing Numbers as (Dual)Resolving and (Dual)Coloring alongside (Dual)Dominating in (Neutrosophic)n-SuperHyperGraph'' in \textbf{Ref.} \cite{HG4} by Henry Garrett (2022),  ``0049 | (Failed)1-Zero-Forcing Number in Neutrosophic Graphs'' in \textbf{Ref.} \cite{HG5} by Henry Garrett (2022),  ``Neutrosophic Failed SuperHyperStable as the Survivors on the Cancer's Neutrosophic Recognition Based on Uncertainty to All Modes in Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG6} by Henry Garrett (2022), ``Extremism of the Attacked Body Under the Cancer's Circumstances Where Cancer's Recognition Titled (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG7} by Henry Garrett (2022), ``(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG8} by Henry Garrett (2022), ``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints'' in \textbf{Ref.} \cite{HG9} by Henry Garrett (2022),  ``Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond'' in \textbf{Ref.} \cite{HG10} by Henry Garrett (2022),  ``(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well- SuperHyperModelled (Neutrosophic) SuperHyperGraphs
'' in \textbf{Ref.} \cite{HG11} by Henry Garrett (2022), ``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints'' in \textbf{Ref.} \cite{HG12} by Henry Garrett (2022), ``Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG13} by Henry Garrett (2022), ``(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances'' in \textbf{Ref.} \cite{HG14} by Henry Garrett (2022), ``(Neutrosophic) SuperHyperAlliances With SuperHyperDefensive and SuperHyperOffensive Type-SuperHyperSet On (Neutrosophic) SuperHyperGraph With (Neutrosophic) SuperHyperModeling of Cancer’s Recognitions And Related (Neutrosophic) SuperHyperClasses'' in \textbf{Ref.} \cite{HG15} by Henry Garrett (2022), ``SuperHyperGirth on SuperHyperGraph and Neutrosophic SuperHyperGraph With SuperHyperModeling of Cancer’s Recognitions'' in \textbf{Ref.} \cite{HG16} by Henry Garrett (2022), ``Some SuperHyperDegrees and Co-SuperHyperDegrees on Neutrosophic SuperHyperGraphs and SuperHyperGraphs Alongside Applications in Cancer’s Treatments'' in \textbf{Ref.} \cite{HG17} by Henry Garrett (2022), ``SuperHyperDominating and SuperHyperResolving on Neutrosophic SuperHyperGraphs And Their Directions in Game Theory and Neutrosophic SuperHyperClasses'' in \textbf{Ref.} \cite{HG18} by Henry Garrett (2022), ``Extreme Failed SuperHyperClique Decides the Failures on the Cancer's Recognition in the Perfect Connections of Cancer's Attacks By SuperHyperModels Named (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG19} by Henry Garrett (2023), ``Indeterminacy On The All Possible Connections of Cells In Front of Cancer's Attacks In The Terms of Neutrosophic Failed SuperHyperClique on Cancer's Recognition called Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG20} by Henry Garrett (2023), ``Perfect Directions Toward Idealism in Cancer's Neutrosophic Recognition Forwarding Neutrosophic SuperHyperClique on Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG21} by Henry Garrett (2023), ``Demonstrating Complete Connections in Every Embedded Regions and Sub-Regions in the Terms of Cancer's Recognition and (Neutrosophic) SuperHyperGraphs With (Neutrosophic) SuperHyperClique'' in \textbf{Ref.} \cite{HG22} by Henry Garrett (2023), ``Different Neutrosophic Types of Neutrosophic Regions titled neutrosophic Failed SuperHyperStable in Cancer’s Neutrosophic Recognition modeled in the Form of Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG23} by Henry Garrett (2023), ``Using the Tool As (Neutrosophic) Failed SuperHyperStable To SuperHyperModel Cancer's Recognition Titled (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG24} by Henry Garrett (2023), ``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints'' in \textbf{Ref.} \cite{HG25} by Henry Garrett (2023), ``(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well-SuperHyperModelled (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG26} by Henry Garrett (2023), ``Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond'' in \textbf{Ref.} \cite{HG27} by Henry Garrett (2022),  ``(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG28} by Henry Garrett (2022), ``Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG29} by Henry Garrett (2022), ``Basic Neutrosophic Notions Concerning SuperHyperDominating and Neutrosophic SuperHyperResolving in SuperHyperGraph'' in \textbf{Ref.} \cite{HG30} by Henry Garrett (2022), ``Initial Material of Neutrosophic Preliminaries to Study Some Neutrosophic Notions Based on Neutrosophic SuperHyperEdge (NSHE) in Neutrosophic SuperHyperGraph (NSHG)''  in \textbf{Ref.} \cite{HG31} by Henry Garrett (2022),  there are some endeavors to formalize the basic SuperHyperNotions about neutrosophic SuperHyperGraph and SuperHyperGraph.    
\\
Some studies and researches about neutrosophic graphs, are proposed as book in \textbf{Ref.} \cite{HG32} by Henry Garrett (2022) which is indexed by Google Scholar and has more than 2347 readers in Scribd. It's titled ``Beyond Neutrosophic Graphs'' and published by Ohio: E-publishing: Educational Publisher 1091 West 1st Ave Grandview Heights, Ohio 43212 United State. This research book covers different types of notions and settings in neutrosophic graph theory and neutrosophic SuperHyperGraph theory.
\\
Also, some studies and researches about neutrosophic graphs, are proposed as book in \textbf{Ref.} \cite{HG33} by Henry Garrett (2022) which is indexed by Google Scholar and has more than 3048 readers in Scribd. It's titled ``Neutrosophic Duality'' and published by Florida: GLOBAL KNOWLEDGE - Publishing House 848 Brickell Ave Ste 950 Miami, Florida 33131 United States. This research book presents different types of notions SuperHyperResolving and SuperHyperDominating in the setting of duality in neutrosophic graph theory and neutrosophic SuperHyperGraph theory. This research book has scrutiny on the complement of the intended set and the intended set, simultaneously. It's smart to consider a set but acting on its complement that what's done in this research book which is popular in the terms of high readers in Scribd.

\section{Motivation and Contributions}
In this research, there are some ideas in the featured frameworks of motivations. I try to bring the motivations in the narrative ways. Some cells have been faced with some attacks from the situation which is caused by the cancer's attacks. In this case, there are some embedded analysis on the ongoing situations which in that, the cells could be labelled as some groups and some groups or individuals have excessive labels which all are raised from the behaviors to overcome the cancer's attacks. In the embedded situations, the individuals of cells and the groups of cells could be considered as ``new groups''. Thus it motivates us to find the proper SuperHyperModels for getting more proper analysis on this messy story. I've found the SuperHyperModels which are officially called ``SuperHyperGraphs'' and ``Neutrosophic SuperHyperGraphs''. In this SuperHyperModel, the cells and the groups of cells are defined as ``SuperHyperVertices'' and the relations between the individuals of cells and the groups of cells are defined as ``SuperHyperEdges''. Thus it's another motivation for us to do research on this SuperHyperModel based on the ``Cancer's Recognition''. Sometimes, the situations get worst. The situation is passed from the certainty and precise style. Thus it's the beyond them. There are three descriptions, namely, the degrees of determinacy, indeterminacy and neutrality, for any object based on vague forms, namely, incomplete data, imprecise data, and uncertain analysis. The latter model could be considered on the previous SuperHyperModel. It's SuperHyperModel. It's SuperHyperGraph but it's officially called ``Neutrosophic SuperHyperGraphs''. The cancer is the disease but the model is going to figure out what's going on this phenomenon. The special case of this disease is considered and as the consequences of the model, some parameters are used. The cells are under attack of this disease but the moves of the cancer in the special region are the matter of mind. The recognition of the cancer could help to find some treatments for this disease. The SuperHyperGraph and neutrosophic SuperHyperGraph are the SuperHyperModels on the ``Cancer's Recognition'' and both bases are the background of this research. Sometimes the cancer has been happened on the region, full of cells, groups of cells and embedded styles. In this segment, the SuperHyperModel proposes some SuperHyperNotions based on the connectivities of the moves of the cancer in the forms of alliances' styles with the formation of the design and the architecture are formally called ``  SuperHyperMatching'' in the themes of jargons and buzzwords. The prefix ``SuperHyper'' refers to the theme of the embedded styles to figure out the background for the SuperHyperNotions.  The recognition of the cancer in the long-term function. The specific region has been assigned by the model [it's called SuperHyperGraph] and the long cycle of the move from the cancer is identified by this research. Sometimes the move of the cancer hasn't be easily identified since there are some determinacy, indeterminacy and neutrality about the moves and the effects of the cancer on that region; this event leads us to choose another model [it's said to be neutrosophic SuperHyperGraph] to have convenient perception on what's happened and what's done. There are some specific models, which are well-known and they've got the names, and some general models. The moves and the traces of the cancer on the complex tracks and between complicated groups of cells could be fantasized by a neutrosophic neutrosophic SuperHyperPath (-/SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel). The aim is to find either the optimal   SuperHyperMatching or the neutrosophic   SuperHyperMatching in those neutrosophic SuperHyperModels. Some general results are introduced. Beyond that in SuperHyperStar, all possible neutrosophic SuperHyperPath s have only two SuperHyperEdges but it's not enough since it's essential to have at least three SuperHyperEdges to form any style of a SuperHyperCycle. There isn't any formation of any SuperHyperCycle but literarily, it's the deformation of any SuperHyperCycle. It, literarily, deforms and it doesn't form.
\begin{question}
How to define the SuperHyperNotions and to do research on them to find the `` amount of   SuperHyperMatching'' of either individual of cells or the groups of cells based on the fixed cell or the fixed group of cells, extensively, the ``amount of   SuperHyperMatching'' based on the fixed groups of cells or the fixed groups of group of cells?
\end{question}
\begin{question}
What are the best descriptions for the ``Cancer's Recognition'' in terms of these messy and dense SuperHyperModels where embedded notions are illustrated?
\end{question}
It's motivation to find notions to use in this dense model is titled ``SuperHyperGraphs''. Thus it motivates us to define different types of ``  SuperHyperMatching'' and ``neutrosophic   SuperHyperMatching'' on ``SuperHyperGraph'' and ``Neutrosophic SuperHyperGraph''. Then the research has taken more motivations to define SuperHyperClasses and to find some connections amid this SuperHyperNotion with other SuperHyperNotions. It motivates us to get some instances and examples to make clarifications about the framework of this research. The general results and some results about some connections are some avenues to make key point of this research, ``Cancer's Recognition'', more understandable and more clear.
\\
The framework of this research is as follows. In the beginning, I introduce basic definitions to clarify about preliminaries. In the subsection ``Preliminaries'', initial definitions about SuperHyperGraphs and neutrosophic SuperHyperGraph are deeply-introduced and in-depth-discussed. The elementary concepts are clarified and illustrated completely and sometimes review literature are applied to make sense about what's going to figure out about the upcoming sections. The main definitions and their clarifications alongside some results about new notions,   SuperHyperMatching and neutrosophic   SuperHyperMatching, are figured out in sections ``  SuperHyperMatching'' and ``Neutrosophic   SuperHyperMatching''. In the sense of tackling on getting results and in order to make sense about continuing the research, the ideas of SuperHyperUniform and Neutrosophic SuperHyperUniform are introduced and as their consequences, corresponded SuperHyperClasses are figured out to debut what's done in this section, titled ``Results on SuperHyperClasses'' and ``Results on Neutrosophic SuperHyperClasses''. As going back to origin of the notions, there are some smart steps toward the common notions to extend the new notions in new frameworks, SuperHyperGraph and Neutrosophic SuperHyperGraph, in the sections ``Results on SuperHyperClasses'' and ``Results on Neutrosophic SuperHyperClasses''. The starter research about the general SuperHyperRelations and as concluding and closing section of theoretical research are contained in the section ``General Results''. Some general SuperHyperRelations are fundamental and they are well-known as fundamental SuperHyperNotions as elicited and discussed in the sections, ``General Results'',
``  SuperHyperMatching'', ``Neutrosophic   SuperHyperMatching'', ``Results on SuperHyperClasses'' and ``Results on Neutrosophic SuperHyperClasses''. There are curious questions about what's done about the SuperHyperNotions to make sense about excellency of this research and going to figure out the word ``best'' as the description and adjective for this research as presented in section, ``  SuperHyperMatching''. The keyword of this research debut in the section ``Applications in Cancer's Recognition'' with two cases and subsections ``Case 1: The Initial Steps Toward SuperHyperBipartite as SuperHyperModel'' and ``Case 2: The Increasing Steps Toward SuperHyperMultipartite as SuperHyperModel''. In the section, ``Open Problems'', there are some scrutiny and discernment on what's done and what's happened in this research in the terms of ``questions'' and ``problems'' to make sense to figure out this research in featured style. The advantages and the limitations of this research alongside about what's done in this research to make sense  and to get sense about what's figured out are included in the section, ``Conclusion and Closing Remarks''.
\section{Preliminaries}
In this subsection, the basic material which is used in this research, is presented. Also, the new ideas and their clarifications are elicited.
\begin{definition}[Neutrosophic Set](\textbf{Ref.}\cite{2},Definition 2.1,p.87).\\
 Let $X$ be a space of points (objects) with generic elements in $X$ denoted by $x;$ then the \textbf{neutrosophic set} $A$ (NS $A$) is an object having the form
$$A = \{< x:T_A(x),I_A(x),F_A(x)>, x\in X\}$$
where the functions $T, I, F: X\rightarrow]^-0,1\leftidx{^+}{[}$ define respectively the a \textbf{truth-membership function}, an \textbf{indeterminacy-membership function}, and a \textbf{falsity-membership function} of the element $x\in X$ to the set $A$ with the condition
$$\leftidx{^-}{0} \leq T_A(x)+I_A(x)+F_A(x)\leq 3^{+}.$$
The functions $T_A(x),I_A(x)$ and $F_A(x)$ are real standard or nonstandard subsets of $]^-0,1\leftidx{^+}{[}.$
\end{definition}
\begin{definition}[Single Valued Neutrosophic Set](\textbf{Ref.}\cite{5},Definition 6,p.2).\\
 Let $X$ be a space of points (objects) with generic elements in $X$ denoted by $x.$ A \textbf{single valued neutrosophic set} $A$ (SVNS $A$) is characterized by truth-membership function $T_A(x),$ an indeterminacy-membership function $I_A(x),$ and a falsity-membership function $F_A(x).$ For each point $x$ in $X,$ $T_A(x),I_A(x),F_A(x)\in [0,1].$ A SVNS $A$ can be written as
 $$A = \{< x:T_A(x),I_A(x),F_A(x)>, x\in X\}.$$
\end{definition}
\begin{definition}
The \textbf{degree of truth-membership}, \textbf{indeterminacy-membership} and \textbf{falsity-membership of the subset} $X\subset A$ of the single valued neutrosophic set  $A = \{< x:T_A(x),I_A(x),F_A(x)>, x\in X\}$:$$T_A(X)=\min[T_A(v_i),T_A(v_j)]_{v_i,v_j\in X},$$
$$I_A(X)=\min[I_A(v_i),I_A(v_j)]_{v_i,v_j\in X},$$
$$ \text{and}~F_A(X)=\min[F_A(v_i),F_A(v_j)]_{v_i,v_j\in X}.$$
\end{definition}
\begin{definition}
The \textbf{support} of $X\subset A$ of the single valued neutrosophic set  $A = \{< x:T_A(x),I_A(x),F_A(x)>, x\in X\}$:
$$supp(X)=\{x\in X:~T_A(x),I_A(x),F_A(x)> 0\}.$$
\end{definition}
\begin{definition}[Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{4},Definition 3,p.291).\\
Assume $V'$ is a given set. A \textbf{neutrosophic SuperHyperGraph} (NSHG) $S$
is an ordered pair $S=(V,E),$ where
\begin{itemize}
\item[$(i)$] $V=\{V_1,V_2,\ldots,V_n\}$ a finite set of finite single valued neutrosophic subsets of $V';$
\item[$(ii)$] $V=\{(V_i,T_{V'}(V_i),I_{V'}(V_i),F_{V'}(V_i)):~T_{V'}(V_i),I_{V'}(V_i),F_{V'}(V_i)\geq0\},~(i=1,2,\ldots,n);$
\item[$(iii)$] $E=\{E_1,E_2,\ldots,E_{n'}\}$ a finite set of finite single valued neutrosophic subsets of $V;$
\item[$(iv)$] $E=\{(E_{i'},T'_{V}(E_{i'}),I'_{V}(E_{i'}),F'_{V}(E_{i'})):~T'_{V}(E_{i'}),I'_{V}(E_{i'}),F'_{V}(E_{i'})\geq0\},~(i'=1,2,\ldots,n');$
\item[$(v)$] $V_i\neq\emptyset,~(i=1,2,\ldots,n);$
\item[$(vi)$] $E_{i'}\neq\emptyset,~(i'=1,2,\ldots,n');$
\item[$(vii)$] $\sum_{i}supp(V_i)=V,~(i=1,2,\ldots,n);$
\item[$(viii)$] $\sum_{i'}supp(E_{i'})=V,~(i'=1,2,\ldots,n');$   
\item[$(ix)$] and the following conditions hold:
$$T'_V(E_{i'})\leq\min[T_{V'}(V_i),T_{V'}(V_j)]_{V_i,V_j\in E_{i'}},$$
$$ I'_V(E_{i'})\leq\min[I_{V'}(V_i),I_{V'}(V_j)]_{V_i,V_j\in E_{i'}},$$
$$ \text{and}~F'_V(E_{i'})\leq\min[F_{V'}(V_i),F_{V'}(V_j)]_{V_i,V_j\in E_{i'}}$$
where $i'=1,2,\ldots,n'.$
\end{itemize}
Here the neutrosophic SuperHyperEdges (NSHE) $E_{j'}$ and the neutrosophic SuperHyperVertices (NSHV) $V_j$ are single valued neutrosophic sets. $T_{V'}(V_i),I_{V'}(V_i),$ and $F_{V'}(V_i)$ denote the degree of truth-membership, the degree of indeterminacy-membership and the degree of falsity-membership the neutrosophic SuperHyperVertex (NSHV) $V_i$ to the neutrosophic SuperHyperVertex (NSHV) $V.$ $T'_{V}(E_{i'}),T'_{V}(E_{i'}),$ and $T'_{V}(E_{i'})$ denote the degree of truth-membership, the degree of indeterminacy-membership and the degree of falsity-membership of the neutrosophic SuperHyperEdge (NSHE) $E_{i'}$ to the neutrosophic SuperHyperEdge (NSHE) $E.$ Thus, the $ii'$th element of the \textbf{incidence matrix} of neutrosophic SuperHyperGraph (NSHG) are of the form $(V_i,T'_{V}(E_{i'}),I'_{V}(E_{i'}),F'_{V}(E_{i'}))$, the sets V and E are crisp sets.
\end{definition}
\begin{definition}[Characterization of the Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{4},Section 4,pp.291-292).\\
 Assume a neutrosophic SuperHyperGraph (NSHG) $S$
is an ordered pair $S=(V,E).$ The neutrosophic SuperHyperEdges (NSHE) $E_{i'}$ and the neutrosophic SuperHyperVertices (NSHV) $V_i$ of neutrosophic SuperHyperGraph (NSHG) $S=(V,E)$ could be characterized as follow-up items.
\begin{itemize}
 \item[$(i)$] If $|V_i|=1,$ then $V_i$ is called \textbf{vertex};
\item[$(ii)$] if $|V_i|\geq1,$ then $V_i$ is called \textbf{SuperVertex};
\item[$(iii)$] if for all $V_i$s are incident in $E_{i'},$ $|V_i|=1,$ and $|E_{i'}|=2,$  then $E_{i'}$ is called \textbf{edge};
\item[$(iv)$] if for all $V_i$s are incident in $E_{i'},$ $|V_i|=1,$ and $|E_{i'}|\geq2,$  then $E_{i'}$ is called \textbf{HyperEdge};
\item[$(v)$] if there's a $V_i$ is incident in $E_{i'}$ such that $|V_i|\geq1,$ and $|E_{i'}|=2,$  then $E_{i'}$ is called \textbf{SuperEdge};
\item[$(vi)$] if there's a $V_i$ is incident in $E_{i'}$ such that $|V_i|\geq1,$ and $|E_{i'}|\geq2,$  then $E_{i'}$ is called \textbf{SuperHyperEdge}.
\end{itemize}
\end{definition}
If we choose different types of binary operations, then we could get hugely diverse types of general forms of neutrosophic SuperHyperGraph (NSHG).
\begin{definition}[t-norm](\textbf{Ref.}\cite{3}, Definition 5.1.1, pp.82-83).\\
 A binary operation $\otimes: [0, 1] \times [0, 1] \rightarrow [0, 1]$ is a \textbf{$t$-norm} if it satisfies the following for $x,y,z,w  \in [0, 1]$:
\begin{itemize}
\item[$(i)$] $1 \otimes x =x;$
\item[$(ii)$] $x \otimes y = y \otimes x;$
\item[$(iii)$] $x \otimes (y \otimes z) = (x \otimes y) \otimes z;$
\item[$(iv)$] If $ w \leq x$ and $y \leq z$ then $w \otimes y \leq x \otimes z.$
\end{itemize}
\end{definition}
\begin{definition}
The \textbf{degree of truth-membership}, \textbf{indeterminacy-membership} and \textbf{falsity-membership of the subset} $X\subset A$ of the single valued neutrosophic set  $A = \{< x:T_A(x),I_A(x),F_A(x)>, x\in X\}$ (with respect to t-norm $T_{norm}$):$$T_A(X)=T_{norm}[T_A(v_i),T_A(v_j)]_{v_i,v_j\in X},$$
$$I_A(X)=T_{norm}[I_A(v_i),I_A(v_j)]_{v_i,v_j\in X},$$
$$ \text{and}~F_A(X)=T_{norm}[F_A(v_i),F_A(v_j)]_{v_i,v_j\in X}.$$
\end{definition}
\begin{definition}
The \textbf{support} of $X\subset A$ of the single valued neutrosophic set  $A = \{< x:T_A(x),I_A(x),F_A(x)>, x\in X\}$:
$$supp(X)=\{x\in X:~T_A(x),I_A(x),F_A(x)> 0\}.$$
\end{definition}
\begin{definition}(General Forms of  Neutrosophic SuperHyperGraph (NSHG)).\\
Assume $V'$ is a given set. A \textbf{neutrosophic SuperHyperGraph} (NSHG) $S$
is an ordered pair $S=(V,E),$ where
\begin{itemize}
\item[$(i)$] $V=\{V_1,V_2,\ldots,V_n\}$ a finite set of finite single valued neutrosophic subsets of $V';$
\item[$(ii)$] $V=\{(V_i,T_{V'}(V_i),I_{V'}(V_i),F_{V'}(V_i)):~T_{V'}(V_i),I_{V'}(V_i),F_{V'}(V_i)\geq0\},~(i=1,2,\ldots,n);$
\item[$(iii)$] $E=\{E_1,E_2,\ldots,E_{n'}\}$ a finite set of finite single valued neutrosophic subsets of $V;$
\item[$(iv)$] $E=\{(E_{i'},T'_{V}(E_{i'}),I'_{V}(E_{i'}),F'_{V}(E_{i'})):~T'_{V}(E_{i'}),I'_{V}(E_{i'}),F'_{V}(E_{i'})\geq0\},~(i'=1,2,\ldots,n');$
\item[$(v)$] $V_i\neq\emptyset,~(i=1,2,\ldots,n);$
\item[$(vi)$] $E_{i'}\neq\emptyset,~(i'=1,2,\ldots,n');$
\item[$(vii)$] $\sum_{i}supp(V_i)=V,~(i=1,2,\ldots,n);$
\item[$(viii)$] $\sum_{i'}supp(E_{i'})=V,~(i'=1,2,\ldots,n').$   
\end{itemize}
Here the neutrosophic SuperHyperEdges (NSHE) $E_{j'}$ and the neutrosophic SuperHyperVertices (NSHV) $V_j$ are single valued neutrosophic sets. $T_{V'}(V_i),I_{V'}(V_i),$ and $F_{V'}(V_i)$ denote the degree of truth-membership, the degree of indeterminacy-membership and the degree of falsity-membership the neutrosophic SuperHyperVertex  (NSHV) $V_i$ to the neutrosophic SuperHyperVertex (NSHV) $V.$ $T'_{V}(E_{i'}),T'_{V}(E_{i'}),$ and $T'_{V}(E_{i'})$ denote the degree of truth-membership, the degree of indeterminacy-membership and the degree of falsity-membership of the neutrosophic SuperHyperEdge (NSHE) $E_{i'}$ to the neutrosophic SuperHyperEdge (NSHE) $E.$ Thus, the $ii'$th element of the \textbf{incidence matrix} of neutrosophic SuperHyperGraph (NSHG) are of the form $(V_i,T'_{V}(E_{i'}),I'_{V}(E_{i'}),F'_{V}(E_{i'}))$, the sets V and E are crisp sets.\end{definition}

\begin{definition}[Characterization of the Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{4},Section 4,pp.291-292).\\
 Assume a neutrosophic SuperHyperGraph (NSHG) $S$
is an ordered pair $S=(V,E).$ The neutrosophic SuperHyperEdges (NSHE) $E_{i'}$ and the neutrosophic SuperHyperVertices (NSHV) $V_i$ of neutrosophic SuperHyperGraph (NSHG) $S=(V,E)$ could be characterized as follow-up items.
\begin{itemize}
 \item[$(i)$] If $|V_i|=1,$ then $V_i$ is called \textbf{vertex};
\item[$(ii)$] if $|V_i|\geq1,$ then $V_i$ is called \textbf{SuperVertex};
\item[$(iii)$] if for all $V_i$s are incident in $E_{i'},$ $|V_i|=1,$ and $|E_{i'}|=2,$  then $E_{i'}$ is called \textbf{edge};
\item[$(iv)$] if for all $V_i$s are incident in $E_{i'},$ $|V_i|=1,$ and $|E_{i'}|\geq2,$  then $E_{i'}$ is called \textbf{HyperEdge};
\item[$(v)$] if there's a $V_i$ is incident in $E_{i'}$ such that $|V_i|\geq1,$ and $|E_{i'}|=2,$  then $E_{i'}$ is called \textbf{SuperEdge};
\item[$(vi)$] if there's a $V_i$ is incident in $E_{i'}$ such that $|V_i|\geq1,$ and $|E_{i'}|\geq2,$  then $E_{i'}$ is called \textbf{SuperHyperEdge}.
\end{itemize}
\end{definition}
This SuperHyperModel is too messy and too dense. Thus there's a need to have some restrictions and conditions on SuperHyperGraph. The special case of this SuperHyperGraph makes the patterns and regularities.
\begin{definition}
 A graph is \textbf{SuperHyperUniform} if it's SuperHyperGraph and the number of elements of SuperHyperEdges are the same.
\end{definition}
To get more visions on    , the some SuperHyperClasses are introduced. It makes to have     more understandable.
\begin{definition}
 Assume a neutrosophic SuperHyperGraph. There are some SuperHyperClasses as follows.
\begin{itemize}
 \item[(i).] It's \textbf{neutrosophic SuperHyperPath } if it's only one SuperVertex as intersection amid two given SuperHyperEdges with two exceptions;
 \item[(ii).] it's \textbf{SuperHyperCycle} if it's only one SuperVertex as intersection amid two given SuperHyperEdges;
  \item[(iii).] it's \textbf{SuperHyperStar} it's only one SuperVertex as intersection amid all SuperHyperEdges;
   \item[(iv).] it's \textbf{SuperHyperBipartite} it's only one SuperVertex as intersection amid two given SuperHyperEdges and these SuperVertices, forming two separate sets, has no SuperHyperEdge in common;
    \item[(v).] it's \textbf{SuperHyperMultiPartite}  it's only one SuperVertex as intersection amid two given SuperHyperEdges and these SuperVertices, forming multi separate sets, has no SuperHyperEdge in common;
     \item[(vi).] it's \textbf{SuperHyperWheel} if it's only one SuperVertex as intersection amid two given SuperHyperEdges and one SuperVertex has one SuperHyperEdge with any common SuperVertex.

\end{itemize}
\end{definition}

\begin{definition}
 Let an ordered pair $S=(V,E)$ be a neutrosophic SuperHyperGraph (NSHG) $S.$ Then a sequence of neutrosophic SuperHyperVertices (NSHV) and neutrosophic SuperHyperEdges (NSHE)
 $$V_1,E_1,V_2,E_2,V_3,\ldots,V_{s-1},E_{s-1},V_s$$
  is called a \textbf{neutrosophic neutrosophic SuperHyperPath } (NSHP) from neutrosophic SuperHyperVertex  (NSHV) $V_1$ to neutrosophic SuperHyperVertex (NSHV) $V_s$ if either of following conditions hold:
  \begin{itemize}
 \item[$(i)$]  $V_i,V_{i+1}\in E_{i'};$
 \item[$(ii)$]   there's a vertex $v_i\in V_i$ such that $v_i,V_{i+1}\in E_{i'};$
  \item[$(iii)$]  there's a SuperVertex $V'_i \in V_i$ such that $V'_i,V_{i+1}\in E_{i'};$
   \item[$(iv)$]   there's a vertex $v_{i+1}\in V_{i+1}$ such that $V_i,v_{i+1}\in E_{i'};$
  \item[$(v)$]  there's a SuperVertex $V'_{i+1} \in V_{i+1}$ such that $V_i,V'_{i+1}\in E_{i'};$
   \item[$(vi)$]   there are a vertex $v_i\in V_i$ and a vertex $v_{i+1}\in V_{i+1}$ such that $v_i,v_{i+1}\in E_{i'};$
  \item[$(vii)$]   there are a vertex $v_i\in V_i$ and a SuperVertex $V'_{i+1} \in V_{i+1}$ such that $v_i,V'_{i+1}\in E_{i'};$  
     \item[$(viii)$]   there are a SuperVertex $V'_i\in V_i$ and a vertex $v_{i+1}\in V_{i+1}$ such that $V'_i,v_{i+1}\in E_{i'};$
  \item[$(ix)$]   there are a SuperVertex $V'_i\in V_i$ and a SuperVertex $V'_{i+1} \in V_{i+1}$ such that $V'_i,V'_{i+1}\in E_{i'}.$  
\end{itemize}
\end{definition}

\begin{definition}(Characterization of the Neutrosophic neutrosophic SuperHyperPath s).\\
 Assume a neutrosophic SuperHyperGraph (NSHG) $S$
is an ordered pair $S=(V,E).$ A neutrosophic neutrosophic SuperHyperPath  (NSHP) from neutrosophic SuperHyperVertex (NSHV) $V_1$ to neutrosophic SuperHyperVertex (NSHV) $V_s$ is sequence of neutrosophic SuperHyperVertices (NSHV) and neutrosophic SuperHyperEdges (NSHE)
 $$V_1,E_1,V_2,E_2,V_3,\ldots,V_{s-1},E_{s-1},V_s,$$
could be characterized as follow-up items.
\begin{itemize}
 \item[$(i)$] If for all $V_i,E_{j'},$ $|V_i|=1,~|E_{j'}|=2,$ then NSHP is called \textbf{path};
\item[$(ii)$] if for all $E_{j'},$ $|E_{j'}|=2,$  and there's $V_i,$ $|V_i|\geq1,$ then NSHP is called \textbf{SuperPath};
\item[$(iii)$] if for all $V_i,E_{j'},$ $|V_i|=1,~|E_{j'}|\geq2,$ then NSHP is called \textbf{HyperPath};
\item[$(iv)$] if there are $V_i,E_{j'},$ $|V_i|\geq1,|E_{j'}|\geq2,$ then NSHP is called \textbf{neutrosophic SuperHyperPath }.
\end{itemize}
\end{definition}
\begin{definition}((neutrosophic) SuperHyperMatching).\\
Assume a SuperHyperGraph.  Then
\begin{itemize}
 \item[$(i)$]
an \textbf{extreme SuperHyperMatching} $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum cardinality of a SuperHyperSet $S$ of high cardinality SuperHyperEdges such that there's no SuperHyperVertex not to in a SuperHyperEdge and there's no SuperHyperEdge to have a SuperHyperVertex in a SuperHyperEdge;
 \item[$(ii)$]
a \textbf{neutrosophic SuperHyperMatching} $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum neutrosophic cardinality of a neutrosophic SuperHyperSet $S$ of high neutrosophic cardinality neutrosophic  SuperHyperEdges such that there's no neutrosophic SuperHyperVertex not to in a neutrosophic SuperHyperEdge and there's no neutrosophic SuperHyperEdge to have a neutrosophic SuperHyperVertex in a neutrosophic SuperHyperEdge;
 \item[$(iii)$]
an \textbf{extreme SuperHyperMatching SuperHyperPolynomial} $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the extreme SuperHyperPolynomial contains the coefficients defined as the number of the maximum cardinality of a SuperHyperSet $S$ of high cardinality SuperHyperEdges such that there's no SuperHyperVertex not to in a SuperHyperEdge and there's no SuperHyperEdge to have a SuperHyperVertex in a SuperHyperEdge and the power is corresponded to its coefficient;
 \item[$(iv)$]
a \textbf{neutrosophic SuperHyperMatching SuperHyperPolynomial} $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the neutrosophic SuperHyperPolynomial contains the neutrosophic coefficients defined as the neutrosophic number of the maximum neutrosophic cardinality of a neutrosophic SuperHyperSet $S$ of high neutrosophic cardinality neutrosophic SuperHyperEdges such that there's no neutrosophic SuperHyperVertex not to in a neutrosophic SuperHyperEdge and there's no neutrosophic SuperHyperEdge to have a neutrosophic SuperHyperVertex in a neutrosophic SuperHyperEdge and the neutrosophic power is neutrosophicly corresponded to its neutrosophic coefficient;
 \item[$(v)$]
an \textbf{extreme R-SuperHyperMatching} $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum cardinality of a SuperHyperSet $S$ of high cardinality SuperHyperVertices such that there's no SuperHyperVertex not to in a SuperHyperEdge and there's no SuperHyperEdge to have a SuperHyperVertex in a SuperHyperEdge;
 \item[$(vi)$]
a \textbf{neutrosophic R-SuperHyperMatching} $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum neutrosophic cardinality of a neutrosophic SuperHyperSet $S$ of high neutrosophic cardinality neutrosophic  SuperHyperVertices such that there's no neutrosophic SuperHyperVertex not to in a neutrosophic SuperHyperEdge and there's no neutrosophic SuperHyperEdge to have a neutrosophic SuperHyperVertex in a neutrosophic SuperHyperEdge;
 \item[$(vii)$]
an \textbf{extreme R-SuperHyperMatching SuperHyperPolynomial} $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the extreme SuperHyperPolynomial contains the coefficients defined as the number of the maximum cardinality of a SuperHyperSet $S$ of high cardinality SuperHyperVertices such that there's no SuperHyperVertex not to in a SuperHyperEdge and there's no SuperHyperEdge to have a SuperHyperVertex in a SuperHyperEdge and the power is corresponded to its coefficient;
 \item[$(viii)$]
a \textbf{neutrosophic R-SuperHyperMatching SuperHyperPolynomial} $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the neutrosophic SuperHyperPolynomial contains the neutrosophic coefficients defined as the neutrosophic number of the maximum neutrosophic cardinality of a neutrosophic SuperHyperSet $S$ of high neutrosophic cardinality  neutrosophic SuperHyperVertices such that there's no neutrosophic SuperHyperVertex not to in a neutrosophic SuperHyperEdge and there's no neutrosophic SuperHyperEdge to have a neutrosophic SuperHyperVertex in a neutrosophic SuperHyperEdge and the neutrosophic power is neutrosophicly corresponded to its neutrosophic coefficient.
\end{itemize}
\end{definition}
\begin{definition}((neutrosophic)$\delta-$SuperHyperMatching).\\
Assume a SuperHyperGraph. Then
\begin{itemize}
 \item[$(i)$] an \textbf{$\delta-$SuperHyperMatching} is a \underline{maximal}     of SuperHyperVertices with a  \underline{maximum} cardinality such that either of the following expressions hold for the (neutrosophic) cardinalities of  SuperHyperNeighbors of $s\in S:$
\begin{eqnarray}
&&|S\cap N(s)| > |S\cap (V\setminus N(s))|+\delta; \label{118EQN1}
 \\&& |S\cap N(s)| < |S\cap (V\setminus N(s))|+\delta. \label{118EQN2}
\end{eqnarray}
The Expression \eqref{118EQN1}, holds if $S$ is an \textbf{$\delta-$SuperHyperOffensive}. And the Expression \eqref{118EQN2}, holds if $S$ is an \textbf{$\delta-$SuperHyperDefensive};
 \item[$(ii)$] a \textbf{neutrosophic $\delta-$SuperHyperMatching} is a \underline{maximal} neutrosophic     of SuperHyperVertices with \underline{maximum} neutrosophic cardinality such that either of the following expressions hold for the neutrosophic cardinalities of  SuperHyperNeighbors of $s\in S:$
\begin{eqnarray}
&&|S\cap N(s)|_{neutrosophic} > |S\cap (V\setminus N(s))|_{neutrosophic}+\delta; \label{118EQN3}
 \\&& |S\cap N(s)|_{neutrosophic} < |S\cap (V\setminus N(s))|_{neutrosophic}+\delta. \label{118EQN4}
\end{eqnarray}
The Expression \eqref{118EQN3}, holds if $S$ is a \textbf{neutrosophic $\delta-$SuperHyperOffensive}. And the Expression \eqref{118EQN4}, holds if $S$ is a \textbf{neutrosophic $\delta-$SuperHyperDefensive}.
\end{itemize}
\end{definition}
For the sake of having a neutrosophic SuperHyperMatching, there's a need to ``\textbf{redefine}'' the notion of ``neutrosophic SuperHyperGraph''. The SuperHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this procedure, there's the usage of the position of labels to assign to the values.
\begin{definition}\label{ 118DEF1}
 Assume a neutrosophic SuperHyperGraph. It's redefined \textbf{neutrosophic SuperHyperGraph} if the Table \eqref{ 118TBL3} holds.
 \begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph Mentioned in the Definition \eqref{ 118DEF1}}
\begin{tabular}[t]{c|c}
\hline
The Values of The Vertices & The Number of Position in Alphabet\\
\hline
The Values of The SuperVertices&The maximum Values of Its Vertices\\
\hline
The Values of The Edges&The maximum Values of Its Vertices\\
\hline
The Values of The HyperEdges&The maximum Values of Its Vertices\\
\hline
The Values of The SuperHyperEdges&The maximum Values of Its Endpoints \\
\hline
\end{tabular}
\label{ 118TBL3}
\end{table}
\end{definition}
It's useful to define a ``neutrosophic'' version of SuperHyperClasses. Since there's more ways to get neutrosophic type-results to make a neutrosophic     more understandable.
\begin{definition}\label{ 118DEF2}
 Assume a neutrosophic SuperHyperGraph. There are some \textbf{neutrosophic SuperHyperClasses} if the Table \eqref{ 118TBL4} holds. Thus  neutrosophic SuperHyperPath ,  SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultiPartite, and   SuperHyperWheel, are
 \textbf{neutrosophic neutrosophic SuperHyperPath },  \textbf{neutrosophic SuperHyperCycle}, \textbf{neutrosophic SuperHyperStar}, \textbf{neutrosophic SuperHyperBipartite}, \textbf{neutrosophic SuperHyperMultiPartite}, and \textbf{neutrosophic SuperHyperWheel} if the Table \eqref{ 118TBL4} holds.
 \begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph, Mentioned in the Definition \eqref{ 118DEF2}}
\begin{tabular}[t]{c|c}
\hline
The Values of The Vertices & The Number of Position in Alphabet\\
\hline
The Values of The SuperVertices&The maximum Values of Its Vertices\\
\hline
The Values of The Edges&The maximum Values of Its Vertices\\
\hline
The Values of The HyperEdges&The maximum Values of Its Vertices\\
\hline
The Values of The SuperHyperEdges&The maximum Values of Its Endpoints \\
\hline
\end{tabular}
\label{ 118TBL4}
\end{table}
\end{definition}
It's useful to define a ``neutrosophic'' version of a SuperHyperMatching. Since there's more ways to get type-results to make a SuperHyperMatching more understandable.
\\
For the sake of having a neutrosophic SuperHyperMatching, there's a need to ``\textbf{redefine}'' the notion of ``   ''. The SuperHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this procedure, there's the usage of the position of labels to assign to the values.
\begin{definition}\label{ 118DEF1}
 Assume a SuperHyperMatching. It's redefined a \textbf{neutrosophic SuperHyperMatching} if the Table \eqref{ 118TBL1} holds.  \begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph Mentioned in the Definition \eqref{ 118DEF1}}
\begin{tabular}[t]{c|c}
\hline
The Values of The Vertices & The Number of Position in Alphabet\\
\hline
The Values of The SuperVertices&The maximum Values of Its Vertices\\
\hline
The Values of The Edges&The maximum Values of Its Vertices\\
\hline
The Values of The HyperEdges&The maximum Values of Its Vertices\\
\hline
The Values of The SuperHyperEdges&The maximum Values of Its Endpoints \\
\hline
\end{tabular}
\label{ 118TBL1}
\end{table}
\end{definition}

\section{Extreme SuperHyperMatching}
The SuperHyperNotion, namely,  SuperHyperMatching, is up. Thus the non-obvious  extreme SuperHyperMatching, $S$ is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: $S$ is  the extreme SuperHyperSet, not: $S$ does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
$S$ in a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$  with a illustrated SuperHyperModeling. It's also, an extreme free-triangle SuperHyperModel. But all only non-obvious simple extreme type-SuperHyperSets of the extreme  SuperHyperMatching amid those obvious[non-obvious] simple extreme type-SuperHyperSets, are $S.$ A connected neutrosophic SuperHyperGraph $ESHG:(V,E)$ as Linearly-over-packed SuperHyperModel is featured on the Figures.
\begin{example}\label{118EXM1}
 Assume the SuperHyperGraphs in the Figures \eqref{118NSHG1}, \eqref{118NSHG2}, \eqref{118NSHG3}, \eqref{118NSHG4}, \eqref{118NSHG5}, \eqref{118NSHG6}, \eqref{118NSHG7}, \eqref{118NSHG8}, \eqref{118NSHG9}, \eqref{118NSHG10}, \eqref{118NSHG11}, \eqref{118NSHG12}, \eqref{118NSHG13}, \eqref{118NSHG14}, \eqref{118NSHG15}, \eqref{118NSHG16}, \eqref{118NSHG17}, \eqref{118NSHG18}, \eqref{118NSHG19}, and \eqref{118NSHG20}.
 \begin{itemize}
 \item On the Figure \eqref{118NSHG1}, the extreme SuperHyperNotion, namely,  extreme SuperHyperMatching, is up.  $E_1$ and $E_3$ are some empty extreme  
 SuperHyperEdges but $E_2$ is a loop extreme SuperHyperEdge and $E_4$ is an extreme SuperHyperEdge. Thus in the terms of extreme SuperHyperNeighbor, there's only one extreme SuperHyperEdge, namely, $E_4.$ The extreme SuperHyperVertex, $V_3$ is extreme isolated means that there's no extreme SuperHyperEdge has it as an extreme endpoint. Thus the extreme SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given  extreme SuperHyperMatching.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
 The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
   is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*} Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}

 \item  On the Figure \eqref{118NSHG2},  the SuperHyperNotion, namely,  SuperHyperMatching, is up.  $E_1$ and $E_3$ SuperHyperMatching are some empty SuperHyperEdges but $E_2$ is a loop SuperHyperEdge and $E_4$ is a SuperHyperEdge. Thus in the terms of SuperHyperNeighbor, there's only one SuperHyperEdge, namely, $E_4.$ The SuperHyperVertex, $V_3$ is isolated means that there's no SuperHyperEdge has it as an endpoint. Thus the extreme SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given  extreme SuperHyperMatching.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
 The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
   is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*} Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}

 \item  On the Figure \eqref{118NSHG3}, the SuperHyperNotion, namely,  SuperHyperMatching, is up.  $E_1,E_2$ and $E_3$ are some empty SuperHyperEdges but $E_4$ is a SuperHyperEdge. Thus in the terms of SuperHyperNeighbor, there's only one SuperHyperEdge, namely, $E_4.$  \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
 The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
   is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*} Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=z~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^3~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
 \item On the Figure \eqref{118NSHG4}, the SuperHyperNotion, namely, a  SuperHyperMatching, is up. There's no empty SuperHyperEdge but $E_3$ are a loop SuperHyperEdge on $\{F\},$ and there are some SuperHyperEdges, namely, $E_1$ on $\{H,V_1,V_3\},$ alongside $E_2$ on $\{O,H,V_4,V_3\}$ and $E_4,E_5$ on $\{N,V_1,V_2,V_3,F\}.$ \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
          \mathcal{C}(NSHG)=\{E_5,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=2z^2~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_3,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^4~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
 The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
          \mathcal{C}(NSHG)=\{E_5,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=2z^2~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_3,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^4~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
          \mathcal{C}(NSHG)=\{E_5,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=2z^2~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_3,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^4~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
   is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
          \mathcal{C}(NSHG)=\{E_5,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=2z^2~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_3,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^4~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
          \mathcal{C}(NSHG)=\{E_5,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=2z^2~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_3,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^4~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*} \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
          \mathcal{C}(NSHG)=\{E_5,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=2z^2~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_3,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^4~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
          \mathcal{C}(NSHG)=\{E_5,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=2z^2~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_3,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^4~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
          \mathcal{C}(NSHG)=\{E_5,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=2z^2~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_3,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^4~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
          \mathcal{C}(NSHG)=\{E_5,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=2z^2~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_3,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^4~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
          \mathcal{C}(NSHG)=\{E_5,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=2z^2~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_3,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^4~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)=\{E_4,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
          \mathcal{C}(NSHG)=\{E_5,E_2\}~\text{is an  extreme SuperHyperMatching.}
 \\&&
 \mathcal{C}(NSHG)=2z^2~\text{is an  extreme SuperHyperMatching SuperHyperPolynomial.}
 \\&&
   \mathcal{C}(NSHG)=\{V_1,V_2,V_3,V_4\}~\text{is an  extreme R-SuperHyperMatching.}
 \\&&
  \mathcal{C}(NSHG)=z^4~{\small\text{is an  extreme R-SuperHyperMatching SuperHyperPolynomial.}}
\end{eqnarray*}
  \item On the Figure \eqref{118NSHG5}, the SuperHyperNotion, namely, SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following extreme SuperHyperSet  of extreme  SuperHyperVertices is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}}=4z^1.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching}}=\{V_1,V_2,V_3,V_4,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching SuperHyperPolynomial}}=2z^5.
\end{eqnarray*}
 The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}}=4z^1.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching}}=\{V_1,V_2,V_3,V_4,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching SuperHyperPolynomial}}=2z^5.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}}=4z^1.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching}}=\{V_1,V_2,V_3,V_4,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching SuperHyperPolynomial}}=2z^5.
\end{eqnarray*}
   is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{are} only \underline{\textbf{same}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{same}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}}=4z^1.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching}}=\{V_1,V_2,V_3,V_4,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching SuperHyperPolynomial}}=2z^5.
\end{eqnarray*}
Doesn't have less than same  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}}=4z^1.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching}}=\{V_1,V_2,V_3,V_4,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching SuperHyperPolynomial}}=2z^5.
\end{eqnarray*}\underline{\textbf{Is}} the obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}}=4z^1.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching}}=\{V_1,V_2,V_3,V_4,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching SuperHyperPolynomial}}=2z^5.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are  only less than same extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}}=4z^1.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching}}=\{V_1,V_2,V_3,V_4,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching SuperHyperPolynomial}}=2z^5.
\end{eqnarray*}
 Thus the obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}}=4z^1.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching}}=\{V_1,V_2,V_3,V_4,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching SuperHyperPolynomial}}=2z^5.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, is: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}}=4z^1.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching}}=\{V_1,V_2,V_3,V_4,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching SuperHyperPolynomial}}=2z^5.
\end{eqnarray*}
Is  the extreme SuperHyperSet, is:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}}=4z^1.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching}}=\{V_1,V_2,V_3,V_4,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching SuperHyperPolynomial}}=2z^5.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}}=4z^1.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching}}=\{V_1,V_2,V_3,V_4,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{\text{Extreme Quasi-R-SuperHyperMatching SuperHyperPolynomial}}=2z^5.
\end{eqnarray*}
In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$ is mentioned as the SuperHyperModel $ESHG:(V,E)$ in the Figure \eqref{118NSHG5}.
 \item  On the Figure \eqref{118NSHG6}, the SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge. \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=5z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}
 The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=5z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=5z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=5z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}  
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=5z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*} \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=5z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}  
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=5z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=5z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}  
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=5z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=5z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}  
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=5z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}

\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=5z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}
   \item On the Figure \eqref{118NSHG7},  the SuperHyperNotion, namely,  SuperHyperMatching, is up.
 The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=2z^7+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=3z^{14}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=2z^7+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=3z^{14}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=2z^7+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=3z^{14}.
\end{eqnarray*}  
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=2z^7+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=3z^{14}.
\end{eqnarray*} \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=2z^7+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=3z^{14}.
\end{eqnarray*}  
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=2z^7+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=3z^{14}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=2z^7+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=3z^{14}.
\end{eqnarray*}  
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=2z^7+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=3z^{14}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=2z^7+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=3z^{14}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=2z^7+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=3z^{14}.
\end{eqnarray*}
   \item On the Figure \eqref{118NSHG8},  the SuperHyperNotion, namely,  SuperHyperMatching, is up. There's neither empty SuperHyperEdge nor loop SuperHyperEdge.  The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
 In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$ of  dense  SuperHyperModel as the Figure \eqref{118NSHG8}.
    \item On the Figure \eqref{118NSHG9}, the SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following extreme SuperHyperSet  of extreme  SuperHyperVertices is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
    \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=3z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}
 The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=3z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=3z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=3z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}  
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=3z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*} \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=3z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}  
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=3z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=3z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}  
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=3z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=3z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=12}^{22}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=3z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{22}.
\end{eqnarray*}
 In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$ of highly-embedding-connected SuperHyperModel as the Figure \eqref{118NSHG9}.
   \item On the Figure \eqref{118NSHG10},  the SuperHyperNotion, namely,  SuperHyperMatching, is up. There's neither empty SuperHyperEdge nor loop SuperHyperEdge.  The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=15}^{17}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=+z^{3}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{14}.
\end{eqnarray*}
In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$ of  dense  SuperHyperModel as the Figure \eqref{118NSHG10}.
   \item On the Figure \eqref{118NSHG11}, the SuperHyperNotion, namely,  SuperHyperMatching, is up. There's neither empty SuperHyperEdge nor loop SuperHyperEdge.  The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
  In a connected extreme SuperHyperGraph $ESHG:(V,E).$
   \item On the Figure \eqref{118NSHG12}, the SuperHyperNotion, namely,  SuperHyperMatching, is up. There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=2}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^5.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{10}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{10}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=2}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^5.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{10}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{10}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=2}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^5.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{10}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{10}.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=2}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^5.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{10}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{10}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=2}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^5.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{10}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{10}.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=2}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^5.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{10}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{10}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=2}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^5.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{10}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{10}.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=2}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^5.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{10}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{10}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=2}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^5.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{10}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{10}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_i\}_{i=2}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^5.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^{10}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^{10}.
\end{eqnarray*}
  In a connected extreme SuperHyperGraph $ESHG:(V,E).$
    \item On the Figure \eqref{118NSHG13}, the SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge.  The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_1,E_3\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3+z^2.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=2z^6.
\end{eqnarray*}  In a connected extreme SuperHyperGraph $ESHG:(V,E).$
  \item On the Figure \eqref{118NSHG14}, the SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_1\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^3.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^2.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_1\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^3.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^2.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_1\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^3.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^2.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_1\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^3.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^2.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_1\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^3.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^2.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_1\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^3.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^2.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_1\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^3.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^2.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_1\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^3.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^2.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_1\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^3.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^2.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_1\}.
 \\&&
    \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^3.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^2.
\end{eqnarray*}
  In a connected extreme SuperHyperGraph $ESHG:(V,E).$  It's noted that this extreme  SuperHyperGraph $ESHG:(V,E)$ is an  extreme  graph $G:(V,E)$ thus the notions in both settings are coincided.
    \item On the Figure \eqref{118NSHG15}, the SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^6.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^6.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^6.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^6.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^6.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^6.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^6.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^6.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^6.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching}=\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-SuperHyperMatching SuperHyperPolynomial}=z^6.
\end{eqnarray*}
 In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$  as Linearly-Connected SuperHyperModel On the Figure \eqref{118NSHG15}.
    \item On the Figure \eqref{118NSHG16}, the SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge.  The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
   \item On the Figure \eqref{118NSHG17}, the SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge.  The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{15}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{15}.
\end{eqnarray*}
 In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$ as Linearly-over-packed SuperHyperModel is featured On the Figure \eqref{118NSHG17}.
   \item On the Figure \eqref{118NSHG18}, the SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^3.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^3.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^3.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
 In a connected neutrosophic SuperHyperGraph $ESHG:(V,E).$
   \item
 On the Figure \eqref{118NSHG19}, the SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^{6}.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{12}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^{6}.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{12}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^{6}.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{12}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^{6}.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{12}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^{6}.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{12}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^{6}.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{12}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^{6}.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{12}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^{6}.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{12}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^{6}.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{12}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i-1}\}_{i=1}^{6}.
 \\&&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_{2i}\}_{i=1}^6.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=2z^{12}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
   \item On the Figure \eqref{118NSHG20}, the SuperHyperNotion, namely,  SuperHyperMatching, is up. The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^6.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^6.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^6.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^6.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^6.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^6.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^6.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^6.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^6.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}=z^6.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
\end{itemize}
\end{example}
 \begin{figure}
 \includegraphics[width=100mm]{118NSHG1.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG1}
\end{figure}
 \begin{figure}
 \includegraphics[width=100mm]{118NSHG2.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG2}
\end{figure}
 \begin{figure}
 \includegraphics[width=100mm]{118NSHG3.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG3}
\end{figure}
 \begin{figure}
 \includegraphics[width=100mm]{118NSHG4.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG4}
\end{figure}
 \begin{figure}
 \includegraphics[width=100mm]{118NSHG5.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG5}
\end{figure}
 \begin{figure}
 \includegraphics[width=100mm]{118NSHG6.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG6}
\end{figure}
 \begin{figure}
 \includegraphics[width=100mm]{118NSHG7.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG7}
\end{figure}
 \begin{figure}
 \includegraphics[width=100mm]{118NSHG8.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG8}
\end{figure}
 \begin{figure}
 \includegraphics[width=100mm]{118NSHG9.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG9}
\end{figure}
 \begin{figure}
 \includegraphics[width=100mm]{118NSHG10.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG10}
\end{figure}
 \begin{figure}
 \includegraphics[width=100mm]{118NSHG11.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG11}
 \end{figure}
  \begin{figure}
 \includegraphics[width=100mm]{118NSHG12.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG12}
\end{figure}
  \begin{figure}
 \includegraphics[width=100mm]{118NSHG13.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG13}
\end{figure}
  \begin{figure}
 \includegraphics[width=100mm]{118NSHG14.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG14}
\end{figure}
  \begin{figure}
 \includegraphics[width=100mm]{118NSHG15.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG15}
\end{figure}
  \begin{figure}
 \includegraphics[width=100mm]{118NSHG16.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG16}
\end{figure}  
\begin{figure}
 \includegraphics[width=100mm]{118NSHG17.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG17}
\end{figure}
  \begin{figure}
 \includegraphics[width=100mm]{118NSHG18.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG18}
\end{figure}
  \begin{figure}
 \includegraphics[width=100mm]{118NSHG19.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG19}
\end{figure}
  \begin{figure}
 \includegraphics[width=100mm]{118NSHG20.png}  
   \caption{The SuperHyperGraphs Associated to the Notions of  SuperHyperMatching in the Example \eqref{118EXM1} }
 \label{118NSHG20}
\end{figure}
\begin{proposition}
 Assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
is an extreme R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of an extreme R-SuperHyperMatching is the cardinality of
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
\end{proposition}
\begin{proof}
  Assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ The SuperHyperSet of the SuperHyperVertices $V\setminus V\setminus \{z\}$ isn't a quasi-R-SuperHyperMatching since neither amount of extreme SuperHyperEdges nor amount of SuperHyperVertices where amount refers to the extreme number of SuperHyperVertices(-/SuperHyperEdges) more than one to form any kind of SuperHyperEdges or any number of SuperHyperEdges. Let us consider the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
This extreme SuperHyperSet of the extreme SuperHyperVertices has the eligibilities to propose property such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices but the maximum extreme cardinality indicates that these extreme  type-SuperHyperSets couldn't give us the extreme lower bound in the term of extreme sharpness. In other words, the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
of the extreme SuperHyperVertices implies at least on-quasi-triangle style is up but sometimes the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
 of the extreme SuperHyperVertices is free-quasi-triangle and it doesn't make a contradiction to the supposition on the connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Thus the minimum case never happens in the generality of the connected loopless neutrosophic SuperHyperGraphs. Thus if we assume in the worst case, literally,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is a quasi-R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of a  quasi-R-SuperHyperMatching is the cardinality of
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 Then we've lost some connected loopless neutrosophic SuperHyperClasses of the connected loopless neutrosophic SuperHyperGraphs titled free-triangle, on-triangle, and their quasi-types but the SuperHyperStable is only up in this quasi-R-SuperHyperMatching. It's the contradiction to that fact on the generality. There are some counterexamples to deny this statement. One of them comes from the setting of the graph titled path and cycle as the counterexamples-classes or reversely direction star as the examples-classes, are well-known classes in that setting and they could be considered as the examples-classes and counterexamples-classes for the tight bound of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  Let $V\setminus V\setminus \{z\}$ in mind. There's no necessity on the SuperHyperEdge since we need at least two SuperHyperVertices to form a SuperHyperEdge. It doesn't withdraw the principles of the main definition since there's no condition to be satisfied but the condition is on the existence of the SuperHyperEdge instead of acting on the SuperHyperVertices. In other words, if there's a SuperHyperEdge, then the extreme SuperHyperSet has the necessary condition for the intended definition to be applied. Thus the $V\setminus V\setminus \{z\}$ is withdrawn not by the conditions of the main definition but by the necessity of the pre-condition on the usage of the main definition.
  \\
  To make sense with the precise words in the terms of ``R-', the follow-up illustrations are coming up.
  \\
  The following extreme SuperHyperSet  of extreme  SuperHyperVertices is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching.
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   The extreme SuperHyperSet of extreme SuperHyperVertices,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperVertices,  
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an \underline{\textbf{extreme R-SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge amid some extreme SuperHyperVertices instead of all given by \underline{\textbf{extreme SuperHyperMatching}} is related to the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   There's   \underline{not} only \underline{\textbf{one}} extreme SuperHyperVertex \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{one}} extreme SuperHyperVertex. But the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   doesn't have less than two  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet since they've come from at least so far an SuperHyperEdge. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperVertices,
 $$(V\setminus V\setminus \{x,z\})\cup\{xy\}$$ or $$(V\setminus V\setminus \{x,z\})\cup\{zy\}$$
   is an  extreme R-SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices instead of all given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme SuperHyperEdge for some amount extreme  SuperHyperVertices instead of all given by that extreme type-SuperHyperSet called the  extreme SuperHyperMatching. There isn't  only less than two extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   Thus the non-obvious  extreme R-SuperHyperMatching,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  is up. The non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is  the extreme SuperHyperSet, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 does includes only less than two SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E)$ but it's impossible in the case, they've corresponded to an SuperHyperEdge. It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  R-SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme R-SuperHyperMatching}},
\end{center}
 is only and only
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$  with a illustrated SuperHyperModeling. It's also, not only an extreme free-triangle embedded SuperHyperModel and an extreme on-triangle embedded SuperHyperModel but also it's an extreme stable embedded SuperHyperModel. But all only non-obvious simple extreme type-SuperHyperSets of the extreme  R-SuperHyperMatching amid those obvious simple extreme type-SuperHyperSets of the extreme SuperHyperMatching, are
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
\\
To sum them up,  assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 is an extreme R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of an extreme  R-SuperHyperMatching is the cardinality of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  \end{proof}
\begin{proposition}
 Assume a simple neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then the extreme number of  R-SuperHyperMatching has, the least cardinality, the lower sharp bound for cardinality, is the extreme cardinality of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
If there's a  R-SuperHyperMatching with the least cardinality, the lower sharp bound for cardinality.
\end{proposition}
\begin{proof}
The extreme structure of the extreme R-SuperHyperMatching decorates the extreme SuperHyperVertices don't have received any extreme connections so as this extreme style implies different versions of extreme SuperHyperEdges with the maximum extreme cardinality in the terms of extreme SuperHyperVertices are spotlight. The lower extreme bound is to have the maximum extreme groups of extreme SuperHyperVertices have perfect extreme connections inside each of SuperHyperEdges and the outside of this extreme SuperHyperSet doesn't matter but regarding the connectedness of the used extreme SuperHyperGraph arising from its extreme properties taken from the fact that it's simple. If there's no more than one extreme SuperHyperVertex in the targeted extreme SuperHyperSet, then there's no extreme connection. Furthermore, the extreme existence of one extreme SuperHyperVertex has no  extreme effect to talk about the extreme R-SuperHyperMatching. Since at least two extreme SuperHyperVertices involve to make a title in the extreme background of the extreme SuperHyperGraph. The extreme SuperHyperGraph is obvious if it has no extreme SuperHyperEdge but at least two extreme SuperHyperVertices make the extreme version of extreme SuperHyperEdge. Thus in the extreme setting of non-obvious extreme SuperHyperGraph, there are at least one extreme SuperHyperEdge. It's necessary to mention that the word ``Simple'' is used as extreme adjective for the initial extreme SuperHyperGraph, induces there's no extreme  appearance of the loop extreme version of the extreme SuperHyperEdge and this extreme SuperHyperGraph is said to be loopless. The extreme adjective ``loop'' on the basic extreme framework engages one extreme SuperHyperVertex but it never happens in this extreme setting. With these extreme bases, on an extreme SuperHyperGraph, there's at least one extreme SuperHyperEdge thus there's at least an extreme R-SuperHyperMatching has the extreme cardinality of an extreme SuperHyperEdge. Thus, an extreme R-SuperHyperMatching has the extreme cardinality at least an extreme SuperHyperEdge. Assume an extreme SuperHyperSet $V\setminus V\setminus \{z\}.$ This extreme SuperHyperSet isn't an extreme R-SuperHyperMatching since either the extreme SuperHyperGraph is an obvious extreme SuperHyperModel thus it never happens since there's no extreme usage of this extreme framework and even more there's no extreme connection inside or the extreme SuperHyperGraph isn't obvious and as its consequences, there's an extreme contradiction with the term ``extreme R-SuperHyperMatching'' since the maximum extreme cardinality never happens for this extreme style of the extreme SuperHyperSet and beyond that there's no extreme connection inside as mentioned in first extreme case in the forms of drawback for this selected extreme SuperHyperSet. Let
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Comes up. This extreme case implies having the extreme style of on-quasi-triangle extreme style on the every extreme elements of this extreme SuperHyperSet. Precisely, the extreme R-SuperHyperMatching is the extreme SuperHyperSet of the extreme SuperHyperVertices such that some extreme amount of the extreme SuperHyperVertices are on-quasi-triangle extreme style. The extreme cardinality of the v SuperHypeSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Is the maximum in comparison to the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But the lower extreme bound is up. Thus the minimum extreme cardinality of the maximum extreme cardinality ends up the extreme discussion. The first extreme term refers to the extreme setting of the extreme SuperHyperGraph but this key point is enough since there's an extreme SuperHyperClass of an extreme SuperHyperGraph has no on-quasi-triangle extreme style amid some amount of its extreme SuperHyperVertices. This extreme setting of the extreme SuperHyperModel proposes an extreme SuperHyperSet has only some amount  extreme SuperHyperVertices from one extreme SuperHyperEdge such that there's no extreme amount of extreme SuperHyperEdges more than one involving these some amount of these extreme SuperHyperVertices. The extreme cardinality of this extreme SuperHyperSet is the maximum and the extreme case is occurred in the minimum extreme situation. To sum them up, the extreme SuperHyperSet
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Has the maximum extreme cardinality such that
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Contains some extreme SuperHyperVertices such that there's distinct-covers-order-amount extreme SuperHyperEdges for amount of extreme SuperHyperVertices taken from the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 It means that the extreme SuperHyperSet of the extreme SuperHyperVertices
  $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an extreme  R-SuperHyperMatching for the extreme SuperHyperGraph as used extreme background in the extreme terms of worst extreme case and the common theme of the lower extreme bound occurred in the specific extreme SuperHyperClasses of the extreme SuperHyperGraphs which are extreme free-quasi-triangle.
  \\
 To make sense with the precise words in the terms of ``R-', the follow-up illustrations are coming up.
  \\
  The following extreme SuperHyperSet  of extreme  SuperHyperVertices is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching.
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   The extreme SuperHyperSet of extreme SuperHyperVertices,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperVertices,  
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an \underline{\textbf{extreme R-SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge amid some extreme SuperHyperVertices instead of all given by \underline{\textbf{extreme SuperHyperMatching}} is related to the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   There's   \underline{not} only \underline{\textbf{one}} extreme SuperHyperVertex \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{one}} extreme SuperHyperVertex. But the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   doesn't have less than two  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet since they've come from at least so far an SuperHyperEdge. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperVertices,
 $$(V\setminus V\setminus \{x,z\})\cup\{xy\}$$ or $$(V\setminus V\setminus \{x,z\})\cup\{zy\}$$
   is an  extreme R-SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices instead of all given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme SuperHyperEdge for some amount extreme  SuperHyperVertices instead of all given by that extreme type-SuperHyperSet called the  extreme SuperHyperMatching. There isn't  only less than two extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   Thus the non-obvious  extreme R-SuperHyperMatching,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  is up. The non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is  the extreme SuperHyperSet, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 does includes only less than two SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E)$ but it's impossible in the case, they've corresponded to an SuperHyperEdge. It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  R-SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme R-SuperHyperMatching}},
\end{center}
 is only and only
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$  with a illustrated SuperHyperModeling. It's also, not only an extreme free-triangle embedded SuperHyperModel and an extreme on-triangle embedded SuperHyperModel but also it's an extreme stable embedded SuperHyperModel. But all only non-obvious simple extreme type-SuperHyperSets of the extreme  R-SuperHyperMatching amid those obvious simple extreme type-SuperHyperSets of the extreme SuperHyperMatching, are
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
\\
To sum them up,  assume a simple neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then the extreme number of  R-SuperHyperMatching has, the least cardinality, the lower sharp bound for cardinality, is the extreme cardinality of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
If there's a  R-SuperHyperMatching with the least cardinality, the lower sharp bound for cardinality.
\end{proof}
\begin{proposition}
Assume a connected extreme  SuperHyperGraph $ESHG:(V,E).$ If an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$ extreme SuperHyperVertices, then the extreme cardinality of the extreme R-SuperHyperMatching is at least
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
 It's straightforward that  the extreme cardinality of the extreme R-SuperHyperMatching is at least the maximum extreme number of extreme SuperHyperVertices of the extreme SuperHyperEdges with the maximum number of the extreme SuperHyperEdges. In other words, the maximum number of the extreme SuperHyperEdges contains the maximum extreme number of extreme SuperHyperVertices are renamed to extreme SuperHyperMatching in some cases but the maximum number of the  extreme SuperHyperEdge with the maximum extreme number of extreme SuperHyperVertices, has the extreme SuperHyperVertices are contained in an extreme R-SuperHyperMatching.
\end{proposition}
\begin{proof}
Assume an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$  extreme number of the extreme SuperHyperVertices. Then every extreme SuperHyperVertex has at least no extreme SuperHyperEdge with others in common. Thus those extreme SuperHyperVertices have the eligibles to be contained in an extreme R-SuperHyperMatching. Those extreme SuperHyperVertices are potentially included in an extreme  style-R-SuperHyperMatching. Formally, consider
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
Are the extreme SuperHyperVertices of an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z.$$ where the $\sim$ isn't an equivalence relation but only the symmetric relation on the extreme  SuperHyperVertices of the extreme SuperHyperGraph. The formal definition is as follows. $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z$$ if and only if $Z_i$ and $Z_j$ are the extreme SuperHyperVertices and there's only and only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ between the extreme SuperHyperVertices $Z_i$ and $Z_j.$ The other definition for the extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ in the terms of extreme R-SuperHyperMatching is
$$\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}.$$
This definition coincides with the definition of the extreme R-SuperHyperMatching but with slightly differences in the maximum extreme cardinality amid those extreme type-SuperHyperSets of the extreme SuperHyperVertices. Thus the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$\max_z|\{Z_1,Z_2,\ldots,Z_z~|~Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}|_{\text{extreme cardinality}},$$
and
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
is formalized with mathematical literatures on the extreme R-SuperHyperMatching. Let $Z_i\overset E\sim Z_j,$ be defined as $Z_i$ and $Z_j$ are the extreme SuperHyperVertices belong to the extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus,  
$$E=\{Z_1,Z_2,\ldots,Z_z~|~Z_i\overset E\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}.$$
Or
$$\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But with the slightly differences,
\begin{eqnarray*}
 &&
 \text{extreme R-SuperHyperMatching}=
 \\&&
 \{Z_1,Z_2,\ldots,Z_z~|~\forall i\neq j,~ i,j=1,2,\ldots,z,~\exists E_x,~Z_i\overset{E_x}\sim Z_j,\}.
\end{eqnarray*}
\begin{eqnarray*}
 &&
 \text{extreme R-SuperHyperMatching}=
 \\&&
V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.
\end{eqnarray*}
Thus $E\in E_{ESHG:(V,E)}$ is an extreme quasi-R-SuperHyperMatching where $E\in E_{ESHG:(V,E)}$ is fixed that means $E_x=E\in E_{ESHG:(V,E)}.$ for all extreme intended SuperHyperVertices but in an extreme SuperHyperMatching, $E_x=E\in E_{ESHG:(V,E)}$ could be different and it's not unique. To sum them up, in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ If an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$ extreme SuperHyperVertices, then the extreme cardinality of the extreme R-SuperHyperMatching is at least
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
 It's straightforward that  the extreme cardinality of the extreme R-SuperHyperMatching is at least the maximum extreme number of extreme SuperHyperVertices of the extreme SuperHyperEdges with the maximum number of the extreme SuperHyperEdges. In other words, the maximum number of the extreme SuperHyperEdges contains the maximum extreme number of extreme SuperHyperVertices are renamed to extreme SuperHyperMatching in some cases but the maximum number of the  extreme SuperHyperEdge with the maximum extreme number of extreme SuperHyperVertices, has the extreme SuperHyperVertices are contained in an extreme R-SuperHyperMatching.
  \\
   To make sense with the precise words in the terms of ``R-', the follow-up illustrations are coming up.
  \\
  The following extreme SuperHyperSet  of extreme  SuperHyperVertices is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching.
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   The extreme SuperHyperSet of extreme SuperHyperVertices,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperVertices,  
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an \underline{\textbf{extreme R-SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge amid some extreme SuperHyperVertices instead of all given by \underline{\textbf{extreme SuperHyperMatching}} is related to the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   There's   \underline{not} only \underline{\textbf{one}} extreme SuperHyperVertex \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{one}} extreme SuperHyperVertex. But the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   doesn't have less than two  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet since they've come from at least so far an SuperHyperEdge. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperVertices,
 $$(V\setminus V\setminus \{x,z\})\cup\{xy\}$$ or $$(V\setminus V\setminus \{x,z\})\cup\{zy\}$$
   is an  extreme R-SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices instead of all given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme SuperHyperEdge for some amount extreme  SuperHyperVertices instead of all given by that extreme type-SuperHyperSet called the  extreme SuperHyperMatching. There isn't  only less than two extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   Thus the non-obvious  extreme R-SuperHyperMatching,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  is up. The non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is  the extreme SuperHyperSet, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 does includes only less than two SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E)$ but it's impossible in the case, they've corresponded to an SuperHyperEdge. It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  R-SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme R-SuperHyperMatching}},
\end{center}
 is only and only
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$  with a illustrated SuperHyperModeling. It's also, not only an extreme free-triangle embedded SuperHyperModel and an extreme on-triangle embedded SuperHyperModel but also it's an extreme stable embedded SuperHyperModel. But all only non-obvious simple extreme type-SuperHyperSets of the extreme  R-SuperHyperMatching amid those obvious simple extreme type-SuperHyperSets of the extreme SuperHyperMatching, are
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
\\
To sum them up,  assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 is an extreme R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of an extreme  R-SuperHyperMatching is the cardinality of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
\\
 To sum them up, in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ If an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$ extreme SuperHyperVertices, then the extreme cardinality of the extreme R-SuperHyperMatching is at least
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
 It's straightforward that  the extreme cardinality of the extreme R-SuperHyperMatching is at least the maximum extreme number of extreme SuperHyperVertices of the extreme SuperHyperEdges with the maximum number of the extreme SuperHyperEdges. In other words, the maximum number of the extreme SuperHyperEdges contains the maximum extreme number of extreme SuperHyperVertices are renamed to extreme SuperHyperMatching in some cases but the maximum number of the  extreme SuperHyperEdge with the maximum extreme number of extreme SuperHyperVertices, has the extreme SuperHyperVertices are contained in an extreme R-SuperHyperMatching.
\end{proof}
\begin{proposition}
Assume a connected non-obvious extreme  SuperHyperGraph $ESHG:(V,E).$ There's only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only the maximum possibilities of the distinct interior extreme SuperHyperVertices inside of any given extreme quasi-R-SuperHyperMatching minus all  extreme SuperHypeNeighbor to some of them but not all of them. In other words, there's only an unique extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only two distinct extreme SuperHyperVertices in an  extreme quasi-R-SuperHyperMatching, minus all  extreme SuperHypeNeighbor to some of them but not all of them.
\end{proposition}
\begin{proof}
The obvious SuperHyperGraph has no extreme SuperHyperEdges. But the non-obvious extreme SuperHyperModel is up. The quasi-SuperHyperModel addresses some issues about the extreme optimal SuperHyperObject. It specially delivers some remarks on the extreme SuperHyperSet of the extreme SuperHyperVertices such that there's distinct amount of extreme SuperHyperEdges for distinct amount of extreme SuperHyperVertices up to all  taken from that extreme SuperHyperSet of the extreme SuperHyperVertices but this extreme SuperHyperSet of the extreme SuperHyperVertices is either has the maximum extreme SuperHyperCardinality or it doesn't have  maximum extreme SuperHyperCardinality. In a non-obvious SuperHyperModel, there's at least one extreme SuperHyperEdge containing at least all extreme SuperHyperVertices. Thus it forms an extreme quasi-R-SuperHyperMatching where the extreme completion of the extreme incidence is up in that.  Thus it's, literarily, an extreme embedded R-SuperHyperMatching. The SuperHyperNotions of embedded SuperHyperSet and quasi-SuperHyperSet coincide. In the original setting, these types of SuperHyperSets only don't satisfy on the maximum SuperHyperCardinality. Thus the embedded setting is elected such that those SuperHyperSets have the maximum extreme SuperHyperCardinality and they're extreme SuperHyperOptimal. The less than two distinct types of extreme SuperHyperVertices are included in the minimum extreme style of the embedded extreme R-SuperHyperMatching. The interior types of the extreme SuperHyperVertices are deciders. Since the extreme number of SuperHyperNeighbors are only  affected by the interior extreme SuperHyperVertices. The common connections, more precise and more formal, the perfect unique connections inside the extreme SuperHyperSet for any distinct types of extreme SuperHyperVertices pose the extreme R-SuperHyperMatching. Thus extreme exterior SuperHyperVertices could be used only in one extreme SuperHyperEdge and in extreme SuperHyperRelation with the interior extreme SuperHyperVertices in that  extreme SuperHyperEdge. In the embedded extreme SuperHyperMatching, there's the usage of exterior extreme SuperHyperVertices since they've more connections inside more than outside. Thus the title ``exterior'' is more relevant than the title ``interior''. One extreme SuperHyperVertex has no connection, inside. Thus, the extreme SuperHyperSet of the extreme SuperHyperVertices with one SuperHyperElement has been ignored in the exploring to lead on the optimal case implying the extreme R-SuperHyperMatching. The extreme R-SuperHyperMatching with the exclusion of the exclusion of all  extreme SuperHyperVertices in one extreme SuperHyperEdge and with other terms, the extreme R-SuperHyperMatching with the inclusion of all extreme SuperHyperVertices in one extreme SuperHyperEdge, is an extreme quasi-R-SuperHyperMatching. To sum them up, in a connected non-obvious extreme  SuperHyperGraph $ESHG:(V,E).$ There's only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only the maximum possibilities of the distinct interior extreme SuperHyperVertices inside of any given extreme quasi-R-SuperHyperMatching minus all  extreme SuperHypeNeighbor to some of them but not all of them. In other words, there's only an unique extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only two distinct extreme SuperHyperVertices in an  extreme quasi-R-SuperHyperMatching, minus all  extreme SuperHypeNeighbor to some of them but not all of them.
  \\
   To make sense with the precise words in the terms of ``R-', the follow-up illustrations are coming up.
  \\
  The following extreme SuperHyperSet  of extreme  SuperHyperVertices is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching.
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   The extreme SuperHyperSet of extreme SuperHyperVertices,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperVertices,  
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an \underline{\textbf{extreme R-SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge amid some extreme SuperHyperVertices instead of all given by \underline{\textbf{extreme SuperHyperMatching}} is related to the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   There's   \underline{not} only \underline{\textbf{one}} extreme SuperHyperVertex \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{one}} extreme SuperHyperVertex. But the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   doesn't have less than two  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet since they've come from at least so far an SuperHyperEdge. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperVertices,
 $$(V\setminus V\setminus \{x,z\})\cup\{xy\}$$ or $$(V\setminus V\setminus \{x,z\})\cup\{zy\}$$
   is an  extreme R-SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices instead of all given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme SuperHyperEdge for some amount extreme  SuperHyperVertices instead of all given by that extreme type-SuperHyperSet called the  extreme SuperHyperMatching. There isn't  only less than two extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   Thus the non-obvious  extreme R-SuperHyperMatching,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  is up. The non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is  the extreme SuperHyperSet, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 does includes only less than two SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E)$ but it's impossible in the case, they've corresponded to an SuperHyperEdge. It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  R-SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme R-SuperHyperMatching}},
\end{center}
 is only and only
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$  with a illustrated SuperHyperModeling. It's also, not only an extreme free-triangle embedded SuperHyperModel and an extreme on-triangle embedded SuperHyperModel but also it's an extreme stable embedded SuperHyperModel. But all only non-obvious simple extreme type-SuperHyperSets of the extreme  R-SuperHyperMatching amid those obvious simple extreme type-SuperHyperSets of the extreme SuperHyperMatching, are
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
\\
To sum them up,  assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 is an extreme R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of an extreme  R-SuperHyperMatching is the cardinality of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
\\
To sum them up, in a connected non-obvious extreme  SuperHyperGraph $ESHG:(V,E).$ There's only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only the maximum possibilities of the distinct interior extreme SuperHyperVertices inside of any given extreme quasi-R-SuperHyperMatching minus all  extreme SuperHypeNeighbor to some of them but not all of them. In other words, there's only an unique extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only two distinct extreme SuperHyperVertices in an  extreme quasi-R-SuperHyperMatching, minus all  extreme SuperHypeNeighbor to some of them but not all of them.
\end{proof}
\begin{proposition}
Assume a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\end{proposition}
\begin{proof}
The main definition of the extreme R-SuperHyperMatching has two titles. An extreme quasi-R-SuperHyperMatching and its corresponded quasi-maximum extreme R-SuperHyperCardinality are two titles in the terms of quasi-R-styles. For any extreme number, there's an extreme quasi-R-SuperHyperMatching with that quasi-maximum extreme SuperHyperCardinality in the terms of the embedded extreme SuperHyperGraph. If there's an embedded extreme SuperHyperGraph, then the extreme quasi-SuperHyperNotions lead us to take the collection of all the extreme quasi-R-SuperHyperMatchings for all extreme numbers less than its extreme corresponded maximum number. The essence of the extreme SuperHyperMatching ends up but this essence starts up in the terms of the extreme quasi-R-SuperHyperMatching, again and more in the operations of collecting all the extreme quasi-R-SuperHyperMatchings acted on the all possible used formations of the extreme SuperHyperGraph to achieve one extreme number. This extreme number is considered as the equivalence class for all corresponded quasi-R-SuperHyperMatchings. Let $z_{\text{Extreme Number}},S_{\text{Extreme SuperHyperSet}}$ and $G_{\text{Extreme SuperHyperMatching}}$ be an extreme number, an extreme SuperHyperSet and an extreme SuperHyperMatching. Then
\begin{eqnarray*}
&&[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}\}.
\end{eqnarray*}
As its consequences, the formal definition of the extreme SuperHyperMatching is re-formalized and redefined as follows.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}\}.
\end{eqnarray*}
To get more precise perceptions, the follow-up expressions propose another formal technical definition for the extreme SuperHyperMatching.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
In more concise and more convenient ways, the modified definition for the extreme SuperHyperMatching poses the upcoming expressions.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
To translate the statement to this mathematical literature, the  formulae will be revised.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
And then,
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
To get more visions in the closer look-up, there's an overall overlook.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Now, the extension of these types of approaches is up. Since the new term, ``extreme SuperHyperNeighborhood'', could be redefined as the collection of the extreme SuperHyperVertices such that any amount of its extreme SuperHyperVertices are incident to an extreme  SuperHyperEdge. It's, literarily,  another name for ``extreme  Quasi-SuperHyperMatching'' but, precisely, it's the generalization of  ``extreme  Quasi-SuperHyperMatching'' since ``extreme Quasi-SuperHyperMatching'' happens ``extreme SuperHyperMatching'' in an extreme SuperHyperGraph as initial framework and background but ``extreme SuperHyperNeighborhood'' may not happens ``extreme SuperHyperMatching'' in an extreme SuperHyperGraph as initial framework and preliminarily background since there are some ambiguities about the extreme SuperHyperCardinality arise from it. To get orderly keywords, the terms, ``extreme SuperHyperNeighborhood'',  ``extreme Quasi-SuperHyperMatching'', and  ``extreme SuperHyperMatching'' are up.
\\
Thus, let $z_{\text{Extreme Number}},N_{\text{Extreme SuperHyperNeighborhood}}$ and $G_{\text{Extreme SuperHyperMatching}}$ be an extreme number, an extreme SuperHyperNeighborhood and an extreme SuperHyperMatching and the new terms are up.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
And with go back to initial structure,
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Thus, in a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\\
  To make sense with the precise words in the terms of ``R-', the follow-up illustrations are coming up.
  \\
  The following extreme SuperHyperSet  of extreme  SuperHyperVertices is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching.
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   The extreme SuperHyperSet of extreme SuperHyperVertices,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperVertices,  
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an \underline{\textbf{extreme R-SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge amid some extreme SuperHyperVertices instead of all given by \underline{\textbf{extreme SuperHyperMatching}} is related to the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   There's   \underline{not} only \underline{\textbf{one}} extreme SuperHyperVertex \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{one}} extreme SuperHyperVertex. But the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   doesn't have less than two  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet since they've come from at least so far an SuperHyperEdge. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperVertices,
 $$(V\setminus V\setminus \{x,z\})\cup\{xy\}$$ or $$(V\setminus V\setminus \{x,z\})\cup\{zy\}$$
   is an  extreme R-SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices instead of all given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme SuperHyperEdge for some amount extreme  SuperHyperVertices instead of all given by that extreme type-SuperHyperSet called the  extreme SuperHyperMatching. There isn't  only less than two extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   Thus the non-obvious  extreme R-SuperHyperMatching,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  is up. The non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is  the extreme SuperHyperSet, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 does includes only less than two SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E)$ but it's impossible in the case, they've corresponded to an SuperHyperEdge. It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  R-SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme R-SuperHyperMatching}},
\end{center}
 is only and only
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$  with a illustrated SuperHyperModeling. It's also, not only an extreme free-triangle embedded SuperHyperModel and an extreme on-triangle embedded SuperHyperModel but also it's an extreme stable embedded SuperHyperModel. But all only non-obvious simple extreme type-SuperHyperSets of the extreme  R-SuperHyperMatching amid those obvious simple extreme type-SuperHyperSets of the extreme SuperHyperMatching, are
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
\\
To sum them up,  assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 is an extreme R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of an extreme  R-SuperHyperMatching is the cardinality of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  \\
To sum them up,  in a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\end{proof}
\begin{proposition}\label{118PRP}
Assume a connected extreme SuperHyperGraph $ESHG:(V,E).$ Any extreme R-SuperHyperMatching only contains all interior extreme SuperHyperVertices and all exterior extreme SuperHyperVertices from the unique extreme SuperHyperEdge where there's any of them has all possible  extreme SuperHyperNeighbors in and there's all  extreme SuperHyperNeighborhoods in with no exception minus all  extreme SuperHypeNeighbors to some of them not all of them  but everything is possible about extreme SuperHyperNeighborhoods and extreme SuperHyperNeighbors out.
\end{proposition}
\begin{proof}
 Assume a connected extreme  SuperHyperGraph $ESHG:(V,E).$ Let an extreme SuperHyperEdge $ESHE: E\in E_{ESHG:(V,E)}$ has some extreme SuperHyperVertices $r.$ Consider all extreme numbers of those extreme SuperHyperVertices from that extreme SuperHyperEdge excluding excluding more than $r$ distinct extreme SuperHyperVertices, exclude to any given extreme SuperHyperSet of the extreme SuperHyperVertices. Consider there's an extreme  R-SuperHyperMatching with the least cardinality, the lower sharp extreme bound for extreme cardinality. Assume a connected extreme SuperHyperGraph $ESHG:(V,E).$ The extreme SuperHyperSet of the extreme SuperHyperVertices $ V_{ESHE}\setminus \{z\}$ is an extreme SuperHyperSet $S$ of  the extreme SuperHyperVertices such that there's an extreme SuperHyperEdge to have  some extreme SuperHyperVertices uniquely but it isn't an extreme R-SuperHyperMatching. Since it doesn't have  \underline{\textbf{the maximum extreme cardinality}} of an extreme SuperHyperSet $S$ of  extreme SuperHyperVertices such that there's an extreme SuperHyperEdge to have some SuperHyperVertices uniquely. The extreme SuperHyperSet of the extreme SuperHyperVertices $V_{ESHE}\cup \{z\}$ is the maximum extreme cardinality of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices but it isn't an extreme R-SuperHyperMatching. Since it \textbf{\underline{doesn't do}} the extreme procedure such that such that there's an extreme SuperHyperEdge to have some extreme  SuperHyperVertices uniquely  [there are at least one extreme SuperHyperVertex outside  implying there's, sometimes in  the connected extreme SuperHyperGraph $ESHG:(V,E),$ an extreme SuperHyperVertex, titled its extreme SuperHyperNeighbor,  to that extreme SuperHyperVertex in the extreme SuperHyperSet $S$ so as $S$ doesn't do ``the extreme procedure''.]. There's  only \textbf{\underline{one}} extreme SuperHyperVertex    \textbf{\underline{outside}} the intended extreme SuperHyperSet, $V_{ESHE}\cup \{z\},$ in the terms of extreme SuperHyperNeighborhood. Thus the obvious extreme R-SuperHyperMatching,  $V_{ESHE}$ is up. The obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching,  $V_{ESHE},$ \textbf{\underline{is}} an extreme SuperHyperSet, $V_{ESHE},$  \textbf{\underline{includes}} only \textbf{\underline{all}}  extreme SuperHyperVertices does forms any kind of extreme pairs are titled   \underline{extreme SuperHyperNeighbors} in a connected extreme SuperHyperGraph $ESHG:(V,E).$ Since the extreme SuperHyperSet of the extreme SuperHyperVertices $V_{ESHE},$ is the \textbf{\underline{maximum extreme SuperHyperCardinality}} of an extreme SuperHyperSet $S$ of  extreme SuperHyperVertices  \textbf{\underline{such that}}  there's an extreme SuperHyperEdge to have some extreme SuperHyperVertices uniquely. Thus, in a connected extreme SuperHyperGraph $ESHG:(V,E).$ Any extreme R-SuperHyperMatching only contains all interior extreme SuperHyperVertices and all exterior extreme SuperHyperVertices from the unique extreme SuperHyperEdge where there's any of them has all possible  extreme SuperHyperNeighbors in and there's all  extreme SuperHyperNeighborhoods in with no exception minus all  extreme SuperHypeNeighbors to some of them not all of them  but everything is possible about extreme SuperHyperNeighborhoods and extreme SuperHyperNeighbors out.
   \\
   To make sense with the precise words in the terms of ``R-', the follow-up illustrations are coming up.
  \\
  The following extreme SuperHyperSet  of extreme  SuperHyperVertices is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching.
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   The extreme SuperHyperSet of extreme SuperHyperVertices,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperVertices,  
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an \underline{\textbf{extreme R-SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge amid some extreme SuperHyperVertices instead of all given by \underline{\textbf{extreme SuperHyperMatching}} is related to the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   There's   \underline{not} only \underline{\textbf{one}} extreme SuperHyperVertex \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{one}} extreme SuperHyperVertex. But the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   doesn't have less than two  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet since they've come from at least so far an SuperHyperEdge. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperVertices,
 $$(V\setminus V\setminus \{x,z\})\cup\{xy\}$$ or $$(V\setminus V\setminus \{x,z\})\cup\{zy\}$$
   is an  extreme R-SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices instead of all given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme SuperHyperEdge for some amount extreme  SuperHyperVertices instead of all given by that extreme type-SuperHyperSet called the  extreme SuperHyperMatching. There isn't  only less than two extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   Thus the non-obvious  extreme R-SuperHyperMatching,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  is up. The non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is  the extreme SuperHyperSet, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 does includes only less than two SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E)$ but it's impossible in the case, they've corresponded to an SuperHyperEdge. It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  R-SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme R-SuperHyperMatching}},
\end{center}
 is only and only
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$  with a illustrated SuperHyperModeling. It's also, not only an extreme free-triangle embedded SuperHyperModel and an extreme on-triangle embedded SuperHyperModel but also it's an extreme stable embedded SuperHyperModel. But all only non-obvious simple extreme type-SuperHyperSets of the extreme  R-SuperHyperMatching amid those obvious simple extreme type-SuperHyperSets of the extreme SuperHyperMatching, are
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
\\
To sum them up,  assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 is an extreme R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of an extreme  R-SuperHyperMatching is the cardinality of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
\\
To sum them up, assume a connected extreme SuperHyperGraph $ESHG:(V,E).$ Any extreme R-SuperHyperMatching only contains all interior extreme SuperHyperVertices and all exterior extreme SuperHyperVertices from the unique extreme SuperHyperEdge where there's any of them has all possible  extreme SuperHyperNeighbors in and there's all  extreme SuperHyperNeighborhoods in with no exception minus all  extreme SuperHypeNeighbors to some of them not all of them  but everything is possible about extreme SuperHyperNeighborhoods and extreme SuperHyperNeighbors out.
 \end{proof}
\begin{remark}
 The words `` extreme SuperHyperMatching'' and ``extreme SuperHyperDominating'' both refer to the maximum extreme type-style. In other words, they  refer to the maximum  extreme SuperHyperNumber and the extreme SuperHyperSet with the maximum extreme SuperHyperCardinality.
\end{remark}
\begin{proposition}
Assume a connected extreme SuperHyperGraph $ESHG:(V,E).$ Consider an extreme  SuperHyperDominating. Then an extreme SuperHyperMatching has the members poses only one extreme representative in an extreme quasi-SuperHyperDominating.
\end{proposition}
\begin{proof}
Assume a connected extreme  SuperHyperGraph $ESHG:(V,E).$ Consider an extreme  SuperHyperDominating. By applying the Proposition \eqref{118PRP}, the extreme results are up. Thus on a connected extreme SuperHyperGraph $ESHG:(V,E).$ Consider an extreme  SuperHyperDominating. Then an extreme SuperHyperMatching has the members poses only one extreme representative in an extreme quasi-SuperHyperDominating.
\end{proof}
\section{Results on Extreme SuperHyperClasses}
The previous extreme approaches apply on the upcoming extreme results on extreme SuperHyperClasses.
\begin{proposition}
Assume a connected extreme SuperHyperPath $ESHP:(V,E).$ Then an extreme quasi-R-SuperHyperMatching-style with the maximum extreme SuperHyperCardinality is an  extreme SuperHyperSet of the interior extreme SuperHyperVertices.
\end{proposition}
\begin{proposition}
Assume a connected  extreme  SuperHyperPath $ESHP:(V,E).$ Then an extreme  quasi-R-SuperHyperMatching is an extreme SuperHyperSet of  the interior extreme SuperHyperVertices with only no extreme exceptions in the form of interior extreme SuperHyperVertices from the unique extreme SuperHyperEdges not excluding only any interior extreme SuperHyperVertices from the extreme unique  SuperHyperEdges. An extreme quasi-R-SuperHyperMatching has the extreme number of all the interior extreme SuperHyperVertices. Also,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
Assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ The SuperHyperSet of the SuperHyperVertices $V\setminus V\setminus \{z\}$ isn't a quasi-R-SuperHyperMatching since neither amount of extreme SuperHyperEdges nor amount of SuperHyperVertices where amount refers to the extreme number of SuperHyperVertices(-/SuperHyperEdges) more than one to form any kind of SuperHyperEdges or any number of SuperHyperEdges. Let us consider the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
This extreme SuperHyperSet of the extreme SuperHyperVertices has the eligibilities to propose property such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices but the maximum extreme cardinality indicates that these extreme  type-SuperHyperSets couldn't give us the extreme lower bound in the term of extreme sharpness. In other words, the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
of the extreme SuperHyperVertices implies at least on-quasi-triangle style is up but sometimes the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
 of the extreme SuperHyperVertices is free-quasi-triangle and it doesn't make a contradiction to the supposition on the connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Thus the minimum case never happens in the generality of the connected loopless neutrosophic SuperHyperGraphs. Thus if we assume in the worst case, literally,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is a quasi-R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of a  quasi-R-SuperHyperMatching is the cardinality of
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 Then we've lost some connected loopless neutrosophic SuperHyperClasses of the connected loopless neutrosophic SuperHyperGraphs titled free-triangle, on-triangle, and their quasi-types but the SuperHyperStable is only up in this quasi-R-SuperHyperMatching. It's the contradiction to that fact on the generality. There are some counterexamples to deny this statement. One of them comes from the setting of the graph titled path and cycle as the counterexamples-classes or reversely direction star as the examples-classes, are well-known classes in that setting and they could be considered as the examples-classes and counterexamples-classes for the tight bound of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  Let $V\setminus V\setminus \{z\}$ in mind. There's no necessity on the SuperHyperEdge since we need at least two SuperHyperVertices to form a SuperHyperEdge. It doesn't withdraw the principles of the main definition since there's no condition to be satisfied but the condition is on the existence of the SuperHyperEdge instead of acting on the SuperHyperVertices. In other words, if there's a SuperHyperEdge, then the extreme SuperHyperSet has the necessary condition for the intended definition to be applied. Thus the $V\setminus V\setminus \{z\}$ is withdrawn not by the conditions of the main definition but by the necessity of the pre-condition on the usage of the main definition.
  \\
  The extreme structure of the extreme R-SuperHyperMatching decorates the extreme SuperHyperVertices don't have received any extreme connections so as this extreme style implies different versions of extreme SuperHyperEdges with the maximum extreme cardinality in the terms of extreme SuperHyperVertices are spotlight. The lower extreme bound is to have the maximum extreme groups of extreme SuperHyperVertices have perfect extreme connections inside each of SuperHyperEdges and the outside of this extreme SuperHyperSet doesn't matter but regarding the connectedness of the used extreme SuperHyperGraph arising from its extreme properties taken from the fact that it's simple. If there's no more than one extreme SuperHyperVertex in the targeted extreme SuperHyperSet, then there's no extreme connection. Furthermore, the extreme existence of one extreme SuperHyperVertex has no  extreme effect to talk about the extreme R-SuperHyperMatching. Since at least two extreme SuperHyperVertices involve to make a title in the extreme background of the extreme SuperHyperGraph. The extreme SuperHyperGraph is obvious if it has no extreme SuperHyperEdge but at least two extreme SuperHyperVertices make the extreme version of extreme SuperHyperEdge. Thus in the extreme setting of non-obvious extreme SuperHyperGraph, there are at least one extreme SuperHyperEdge. It's necessary to mention that the word ``Simple'' is used as extreme adjective for the initial extreme SuperHyperGraph, induces there's no extreme  appearance of the loop extreme version of the extreme SuperHyperEdge and this extreme SuperHyperGraph is said to be loopless. The extreme adjective ``loop'' on the basic extreme framework engages one extreme SuperHyperVertex but it never happens in this extreme setting. With these extreme bases, on an extreme SuperHyperGraph, there's at least one extreme SuperHyperEdge thus there's at least an extreme R-SuperHyperMatching has the extreme cardinality of an extreme SuperHyperEdge. Thus, an extreme R-SuperHyperMatching has the extreme cardinality at least an extreme SuperHyperEdge. Assume an extreme SuperHyperSet $V\setminus V\setminus \{z\}.$ This extreme SuperHyperSet isn't an extreme R-SuperHyperMatching since either the extreme SuperHyperGraph is an obvious extreme SuperHyperModel thus it never happens since there's no extreme usage of this extreme framework and even more there's no extreme connection inside or the extreme SuperHyperGraph isn't obvious and as its consequences, there's an extreme contradiction with the term ``extreme R-SuperHyperMatching'' since the maximum extreme cardinality never happens for this extreme style of the extreme SuperHyperSet and beyond that there's no extreme connection inside as mentioned in first extreme case in the forms of drawback for this selected extreme SuperHyperSet. Let
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Comes up. This extreme case implies having the extreme style of on-quasi-triangle extreme style on the every extreme elements of this extreme SuperHyperSet. Precisely, the extreme R-SuperHyperMatching is the extreme SuperHyperSet of the extreme SuperHyperVertices such that some extreme amount of the extreme SuperHyperVertices are on-quasi-triangle extreme style. The extreme cardinality of the v SuperHypeSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Is the maximum in comparison to the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But the lower extreme bound is up. Thus the minimum extreme cardinality of the maximum extreme cardinality ends up the extreme discussion. The first extreme term refers to the extreme setting of the extreme SuperHyperGraph but this key point is enough since there's an extreme SuperHyperClass of an extreme SuperHyperGraph has no on-quasi-triangle extreme style amid some amount of its extreme SuperHyperVertices. This extreme setting of the extreme SuperHyperModel proposes an extreme SuperHyperSet has only some amount  extreme SuperHyperVertices from one extreme SuperHyperEdge such that there's no extreme amount of extreme SuperHyperEdges more than one involving these some amount of these extreme SuperHyperVertices. The extreme cardinality of this extreme SuperHyperSet is the maximum and the extreme case is occurred in the minimum extreme situation. To sum them up, the extreme SuperHyperSet
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Has the maximum extreme cardinality such that
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Contains some extreme SuperHyperVertices such that there's distinct-covers-order-amount extreme SuperHyperEdges for amount of extreme SuperHyperVertices taken from the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 It means that the extreme SuperHyperSet of the extreme SuperHyperVertices
  $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an extreme  R-SuperHyperMatching for the extreme SuperHyperGraph as used extreme background in the extreme terms of worst extreme case and the common theme of the lower extreme bound occurred in the specific extreme SuperHyperClasses of the extreme SuperHyperGraphs which are extreme free-quasi-triangle.
  \\
Assume an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$  extreme number of the extreme SuperHyperVertices. Then every extreme SuperHyperVertex has at least no extreme SuperHyperEdge with others in common. Thus those extreme SuperHyperVertices have the eligibles to be contained in an extreme R-SuperHyperMatching. Those extreme SuperHyperVertices are potentially included in an extreme  style-R-SuperHyperMatching. Formally, consider
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
Are the extreme SuperHyperVertices of an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z.$$ where the $\sim$ isn't an equivalence relation but only the symmetric relation on the extreme  SuperHyperVertices of the extreme SuperHyperGraph. The formal definition is as follows. $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z$$ if and only if $Z_i$ and $Z_j$ are the extreme SuperHyperVertices and there's only and only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ between the extreme SuperHyperVertices $Z_i$ and $Z_j.$ The other definition for the extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ in the terms of extreme R-SuperHyperMatching is
$$\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}.$$
This definition coincides with the definition of the extreme R-SuperHyperMatching but with slightly differences in the maximum extreme cardinality amid those extreme type-SuperHyperSets of the extreme SuperHyperVertices. Thus the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$\max_z|\{Z_1,Z_2,\ldots,Z_z~|~Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}|_{\text{extreme cardinality}},$$
and
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
is formalized with mathematical literatures on the extreme R-SuperHyperMatching. Let $Z_i\overset E\sim Z_j,$ be defined as $Z_i$ and $Z_j$ are the extreme SuperHyperVertices belong to the extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus,  
$$E=\{Z_1,Z_2,\ldots,Z_z~|~Z_i\overset E\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}.$$
Or
$$\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But with the slightly differences,
\begin{eqnarray*}
 &&
 \text{extreme R-SuperHyperMatching}=
 \\&&
 \{Z_1,Z_2,\ldots,Z_z~|~\forall i\neq j,~ i,j=1,2,\ldots,z,~\exists E_x,~Z_i\overset{E_x}\sim Z_j,\}.
\end{eqnarray*}
\begin{eqnarray*}
 &&
 \text{extreme R-SuperHyperMatching}=
 \\&&
V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.
\end{eqnarray*}
Thus $E\in E_{ESHG:(V,E)}$ is an extreme quasi-R-SuperHyperMatching where $E\in E_{ESHG:(V,E)}$ is fixed that means $E_x=E\in E_{ESHG:(V,E)}.$ for all extreme intended SuperHyperVertices but in an extreme SuperHyperMatching, $E_x=E\in E_{ESHG:(V,E)}$ could be different and it's not unique. To sum them up, in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ If an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$ extreme SuperHyperVertices, then the extreme cardinality of the extreme R-SuperHyperMatching is at least
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
 It's straightforward that  the extreme cardinality of the extreme R-SuperHyperMatching is at least the maximum extreme number of extreme SuperHyperVertices of the extreme SuperHyperEdges with the maximum number of the extreme SuperHyperEdges. In other words, the maximum number of the extreme SuperHyperEdges contains the maximum extreme number of extreme SuperHyperVertices are renamed to extreme SuperHyperMatching in some cases but the maximum number of the  extreme SuperHyperEdge with the maximum extreme number of extreme SuperHyperVertices, has the extreme SuperHyperVertices are contained in an extreme R-SuperHyperMatching.
\\
The obvious SuperHyperGraph has no extreme SuperHyperEdges. But the non-obvious extreme SuperHyperModel is up. The quasi-SuperHyperModel addresses some issues about the extreme optimal SuperHyperObject. It specially delivers some remarks on the extreme SuperHyperSet of the extreme SuperHyperVertices such that there's distinct amount of extreme SuperHyperEdges for distinct amount of extreme SuperHyperVertices up to all  taken from that extreme SuperHyperSet of the extreme SuperHyperVertices but this extreme SuperHyperSet of the extreme SuperHyperVertices is either has the maximum extreme SuperHyperCardinality or it doesn't have  maximum extreme SuperHyperCardinality. In a non-obvious SuperHyperModel, there's at least one extreme SuperHyperEdge containing at least all extreme SuperHyperVertices. Thus it forms an extreme quasi-R-SuperHyperMatching where the extreme completion of the extreme incidence is up in that.  Thus it's, literarily, an extreme embedded R-SuperHyperMatching. The SuperHyperNotions of embedded SuperHyperSet and quasi-SuperHyperSet coincide. In the original setting, these types of SuperHyperSets only don't satisfy on the maximum SuperHyperCardinality. Thus the embedded setting is elected such that those SuperHyperSets have the maximum extreme SuperHyperCardinality and they're extreme SuperHyperOptimal. The less than two distinct types of extreme SuperHyperVertices are included in the minimum extreme style of the embedded extreme R-SuperHyperMatching. The interior types of the extreme SuperHyperVertices are deciders. Since the extreme number of SuperHyperNeighbors are only  affected by the interior extreme SuperHyperVertices. The common connections, more precise and more formal, the perfect unique connections inside the extreme SuperHyperSet for any distinct types of extreme SuperHyperVertices pose the extreme R-SuperHyperMatching. Thus extreme exterior SuperHyperVertices could be used only in one extreme SuperHyperEdge and in extreme SuperHyperRelation with the interior extreme SuperHyperVertices in that  extreme SuperHyperEdge. In the embedded extreme SuperHyperMatching, there's the usage of exterior extreme SuperHyperVertices since they've more connections inside more than outside. Thus the title ``exterior'' is more relevant than the title ``interior''. One extreme SuperHyperVertex has no connection, inside. Thus, the extreme SuperHyperSet of the extreme SuperHyperVertices with one SuperHyperElement has been ignored in the exploring to lead on the optimal case implying the extreme R-SuperHyperMatching. The extreme R-SuperHyperMatching with the exclusion of the exclusion of all  extreme SuperHyperVertices in one extreme SuperHyperEdge and with other terms, the extreme R-SuperHyperMatching with the inclusion of all extreme SuperHyperVertices in one extreme SuperHyperEdge, is an extreme quasi-R-SuperHyperMatching. To sum them up, in a connected non-obvious extreme  SuperHyperGraph $ESHG:(V,E).$ There's only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only the maximum possibilities of the distinct interior extreme SuperHyperVertices inside of any given extreme quasi-R-SuperHyperMatching minus all  extreme SuperHypeNeighbor to some of them but not all of them. In other words, there's only an unique extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only two distinct extreme SuperHyperVertices in an  extreme quasi-R-SuperHyperMatching, minus all  extreme SuperHypeNeighbor to some of them but not all of them.
\\
The main definition of the extreme R-SuperHyperMatching has two titles. An extreme quasi-R-SuperHyperMatching and its corresponded quasi-maximum extreme R-SuperHyperCardinality are two titles in the terms of quasi-R-styles. For any extreme number, there's an extreme quasi-R-SuperHyperMatching with that quasi-maximum extreme SuperHyperCardinality in the terms of the embedded extreme SuperHyperGraph. If there's an embedded extreme SuperHyperGraph, then the extreme quasi-SuperHyperNotions lead us to take the collection of all the extreme quasi-R-SuperHyperMatchings for all extreme numbers less than its extreme corresponded maximum number. The essence of the extreme SuperHyperMatching ends up but this essence starts up in the terms of the extreme quasi-R-SuperHyperMatching, again and more in the operations of collecting all the extreme quasi-R-SuperHyperMatchings acted on the all possible used formations of the extreme SuperHyperGraph to achieve one extreme number. This extreme number is considered as the equivalence class for all corresponded quasi-R-SuperHyperMatchings. Let $z_{\text{Extreme Number}},S_{\text{Extreme SuperHyperSet}}$ and $G_{\text{Extreme SuperHyperMatching}}$ be an extreme number, an extreme SuperHyperSet and an extreme SuperHyperMatching. Then
\begin{eqnarray*}
&&[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}\}.
\end{eqnarray*}
As its consequences, the formal definition of the extreme SuperHyperMatching is re-formalized and redefined as follows.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}\}.
\end{eqnarray*}
To get more precise perceptions, the follow-up expressions propose another formal technical definition for the extreme SuperHyperMatching.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
In more concise and more convenient ways, the modified definition for the extreme SuperHyperMatching poses the upcoming expressions.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
To translate the statement to this mathematical literature, the  formulae will be revised.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
And then,
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
To get more visions in the closer look-up, there's an overall overlook.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Now, the extension of these types of approaches is up. Since the new term, ``extreme SuperHyperNeighborhood'', could be redefined as the collection of the extreme SuperHyperVertices such that any amount of its extreme SuperHyperVertices are incident to an extreme  SuperHyperEdge. It's, literarily,  another name for ``extreme  Quasi-SuperHyperMatching'' but, precisely, it's the generalization of  ``extreme  Quasi-SuperHyperMatching'' since ``extreme Quasi-SuperHyperMatching'' happens ``extreme SuperHyperMatching'' in an extreme SuperHyperGraph as initial framework and background but ``extreme SuperHyperNeighborhood'' may not happens ``extreme SuperHyperMatching'' in an extreme SuperHyperGraph as initial framework and preliminarily background since there are some ambiguities about the extreme SuperHyperCardinality arise from it. To get orderly keywords, the terms, ``extreme SuperHyperNeighborhood'',  ``extreme Quasi-SuperHyperMatching'', and  ``extreme SuperHyperMatching'' are up.
\\
Thus, let $z_{\text{Extreme Number}},N_{\text{Extreme SuperHyperNeighborhood}}$ and $G_{\text{Extreme SuperHyperMatching}}$ be an extreme number, an extreme SuperHyperNeighborhood and an extreme SuperHyperMatching and the new terms are up.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
And with go back to initial structure,
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Thus, in a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\\
  To make sense with the precise words in the terms of ``R-', the follow-up illustrations are coming up.
  \\
  The following extreme SuperHyperSet  of extreme  SuperHyperVertices is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching.
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   The extreme SuperHyperSet of extreme SuperHyperVertices,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperVertices,  
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an \underline{\textbf{extreme R-SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge amid some extreme SuperHyperVertices instead of all given by \underline{\textbf{extreme SuperHyperMatching}} is related to the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   There's   \underline{not} only \underline{\textbf{one}} extreme SuperHyperVertex \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{one}} extreme SuperHyperVertex. But the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   doesn't have less than two  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet since they've come from at least so far an SuperHyperEdge. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperVertices,
 $$(V\setminus V\setminus \{x,z\})\cup\{xy\}$$ or $$(V\setminus V\setminus \{x,z\})\cup\{zy\}$$
   is an  extreme R-SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices instead of all given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme SuperHyperEdge for some amount extreme  SuperHyperVertices instead of all given by that extreme type-SuperHyperSet called the  extreme SuperHyperMatching. There isn't  only less than two extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   Thus the non-obvious  extreme R-SuperHyperMatching,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  is up. The non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is  the extreme SuperHyperSet, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 does includes only less than two SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E)$ but it's impossible in the case, they've corresponded to an SuperHyperEdge. It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  R-SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme R-SuperHyperMatching}},
\end{center}
 is only and only
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$  with a illustrated SuperHyperModeling. It's also, not only an extreme free-triangle embedded SuperHyperModel and an extreme on-triangle embedded SuperHyperModel but also it's an extreme stable embedded SuperHyperModel. But all only non-obvious simple extreme type-SuperHyperSets of the extreme  R-SuperHyperMatching amid those obvious simple extreme type-SuperHyperSets of the extreme SuperHyperMatching, are
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
\\
To sum them up,  assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 is an extreme R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of an extreme  R-SuperHyperMatching is the cardinality of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  \\
To sum them up,  in a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\\
Assume a connected extreme  SuperHyperGraph $ESHG:(V,E).$ Let an extreme SuperHyperEdge $ESHE: E\in E_{ESHG:(V,E)}$ has some extreme SuperHyperVertices $r.$ Consider all extreme numbers of those extreme SuperHyperVertices from that extreme SuperHyperEdge excluding excluding more than $r$ distinct extreme SuperHyperVertices, exclude to any given extreme SuperHyperSet of the extreme SuperHyperVertices. Consider there's an extreme  R-SuperHyperMatching with the least cardinality, the lower sharp extreme bound for extreme cardinality. Assume a connected extreme SuperHyperGraph $ESHG:(V,E).$ The extreme SuperHyperSet of the extreme SuperHyperVertices $ V_{ESHE}\setminus \{z\}$ is an extreme SuperHyperSet $S$ of  the extreme SuperHyperVertices such that there's an extreme SuperHyperEdge to have  some extreme SuperHyperVertices uniquely but it isn't an extreme R-SuperHyperMatching. Since it doesn't have  \underline{\textbf{the maximum extreme cardinality}} of an extreme SuperHyperSet $S$ of  extreme SuperHyperVertices such that there's an extreme SuperHyperEdge to have some SuperHyperVertices uniquely. The extreme SuperHyperSet of the extreme SuperHyperVertices $V_{ESHE}\cup \{z\}$ is the maximum extreme cardinality of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices but it isn't an extreme R-SuperHyperMatching. Since it \textbf{\underline{doesn't do}} the extreme procedure such that such that there's an extreme SuperHyperEdge to have some extreme  SuperHyperVertices uniquely  [there are at least one extreme SuperHyperVertex outside  implying there's, sometimes in  the connected extreme SuperHyperGraph $ESHG:(V,E),$ an extreme SuperHyperVertex, titled its extreme SuperHyperNeighbor,  to that extreme SuperHyperVertex in the extreme SuperHyperSet $S$ so as $S$ doesn't do ``the extreme procedure''.]. There's  only \textbf{\underline{one}} extreme SuperHyperVertex    \textbf{\underline{outside}} the intended extreme SuperHyperSet, $V_{ESHE}\cup \{z\},$ in the terms of extreme SuperHyperNeighborhood. Thus the obvious extreme R-SuperHyperMatching,  $V_{ESHE}$ is up. The obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching,  $V_{ESHE},$ \textbf{\underline{is}} an extreme SuperHyperSet, $V_{ESHE},$  \textbf{\underline{includes}} only \textbf{\underline{all}}  extreme SuperHyperVertices does forms any kind of extreme pairs are titled   \underline{extreme SuperHyperNeighbors} in a connected extreme SuperHyperGraph $ESHG:(V,E).$ Since the extreme SuperHyperSet of the extreme SuperHyperVertices $V_{ESHE},$ is the \textbf{\underline{maximum extreme SuperHyperCardinality}} of an extreme SuperHyperSet $S$ of  extreme SuperHyperVertices  \textbf{\underline{such that}}  there's an extreme SuperHyperEdge to have some extreme SuperHyperVertices uniquely. Thus, in a connected extreme SuperHyperGraph $ESHG:(V,E).$ Any extreme R-SuperHyperMatching only contains all interior extreme SuperHyperVertices and all exterior extreme SuperHyperVertices from the unique extreme SuperHyperEdge where there's any of them has all possible  extreme SuperHyperNeighbors in and there's all  extreme SuperHyperNeighborhoods in with no exception minus all  extreme SuperHypeNeighbors to some of them not all of them  but everything is possible about extreme SuperHyperNeighborhoods and extreme SuperHyperNeighbors out.
\\
The SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
 \end{proof}
 \begin{example}\label{118EXM18a}
In the Figure \eqref{118NSHG18a}, the connected extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The extreme SuperHyperSet,  in the extreme SuperHyperModel \eqref{118NSHG18a}, is the  SuperHyperMatching.
\begin{figure}
\includegraphics[width=100mm]{118NSHG18.png}
 \caption{An extreme SuperHyperPath Associated to the Notions of  extreme SuperHyperMatching in the Example \eqref{118EXM18a}}
 \label{118NSHG18a}
\end{figure}
\end{example}

\begin{proposition}
Assume a connected   extreme SuperHyperCycle $ESHC:(V,E).$ Then an extreme  quasi-R-SuperHyperMatching is an extreme SuperHyperSet of the interior extreme SuperHyperVertices with only no extreme exceptions on  the form of interior extreme SuperHyperVertices from the same extreme SuperHyperNeighborhoods not excluding any extreme SuperHyperVertex. An extreme quasi-R-SuperHyperMatching has the extreme half number of all the  extreme SuperHyperEdges in the terms of the maximum extreme cardinality.  Also,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
Assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ The SuperHyperSet of the SuperHyperVertices $V\setminus V\setminus \{z\}$ isn't a quasi-R-SuperHyperMatching since neither amount of extreme SuperHyperEdges nor amount of SuperHyperVertices where amount refers to the extreme number of SuperHyperVertices(-/SuperHyperEdges) more than one to form any kind of SuperHyperEdges or any number of SuperHyperEdges. Let us consider the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
This extreme SuperHyperSet of the extreme SuperHyperVertices has the eligibilities to propose property such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices but the maximum extreme cardinality indicates that these extreme  type-SuperHyperSets couldn't give us the extreme lower bound in the term of extreme sharpness. In other words, the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
of the extreme SuperHyperVertices implies at least on-quasi-triangle style is up but sometimes the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
 of the extreme SuperHyperVertices is free-quasi-triangle and it doesn't make a contradiction to the supposition on the connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Thus the minimum case never happens in the generality of the connected loopless neutrosophic SuperHyperGraphs. Thus if we assume in the worst case, literally,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is a quasi-R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of a  quasi-R-SuperHyperMatching is the cardinality of
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 Then we've lost some connected loopless neutrosophic SuperHyperClasses of the connected loopless neutrosophic SuperHyperGraphs titled free-triangle, on-triangle, and their quasi-types but the SuperHyperStable is only up in this quasi-R-SuperHyperMatching. It's the contradiction to that fact on the generality. There are some counterexamples to deny this statement. One of them comes from the setting of the graph titled path and cycle as the counterexamples-classes or reversely direction star as the examples-classes, are well-known classes in that setting and they could be considered as the examples-classes and counterexamples-classes for the tight bound of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  Let $V\setminus V\setminus \{z\}$ in mind. There's no necessity on the SuperHyperEdge since we need at least two SuperHyperVertices to form a SuperHyperEdge. It doesn't withdraw the principles of the main definition since there's no condition to be satisfied but the condition is on the existence of the SuperHyperEdge instead of acting on the SuperHyperVertices. In other words, if there's a SuperHyperEdge, then the extreme SuperHyperSet has the necessary condition for the intended definition to be applied. Thus the $V\setminus V\setminus \{z\}$ is withdrawn not by the conditions of the main definition but by the necessity of the pre-condition on the usage of the main definition.
  \\
  The extreme structure of the extreme R-SuperHyperMatching decorates the extreme SuperHyperVertices don't have received any extreme connections so as this extreme style implies different versions of extreme SuperHyperEdges with the maximum extreme cardinality in the terms of extreme SuperHyperVertices are spotlight. The lower extreme bound is to have the maximum extreme groups of extreme SuperHyperVertices have perfect extreme connections inside each of SuperHyperEdges and the outside of this extreme SuperHyperSet doesn't matter but regarding the connectedness of the used extreme SuperHyperGraph arising from its extreme properties taken from the fact that it's simple. If there's no more than one extreme SuperHyperVertex in the targeted extreme SuperHyperSet, then there's no extreme connection. Furthermore, the extreme existence of one extreme SuperHyperVertex has no  extreme effect to talk about the extreme R-SuperHyperMatching. Since at least two extreme SuperHyperVertices involve to make a title in the extreme background of the extreme SuperHyperGraph. The extreme SuperHyperGraph is obvious if it has no extreme SuperHyperEdge but at least two extreme SuperHyperVertices make the extreme version of extreme SuperHyperEdge. Thus in the extreme setting of non-obvious extreme SuperHyperGraph, there are at least one extreme SuperHyperEdge. It's necessary to mention that the word ``Simple'' is used as extreme adjective for the initial extreme SuperHyperGraph, induces there's no extreme  appearance of the loop extreme version of the extreme SuperHyperEdge and this extreme SuperHyperGraph is said to be loopless. The extreme adjective ``loop'' on the basic extreme framework engages one extreme SuperHyperVertex but it never happens in this extreme setting. With these extreme bases, on an extreme SuperHyperGraph, there's at least one extreme SuperHyperEdge thus there's at least an extreme R-SuperHyperMatching has the extreme cardinality of an extreme SuperHyperEdge. Thus, an extreme R-SuperHyperMatching has the extreme cardinality at least an extreme SuperHyperEdge. Assume an extreme SuperHyperSet $V\setminus V\setminus \{z\}.$ This extreme SuperHyperSet isn't an extreme R-SuperHyperMatching since either the extreme SuperHyperGraph is an obvious extreme SuperHyperModel thus it never happens since there's no extreme usage of this extreme framework and even more there's no extreme connection inside or the extreme SuperHyperGraph isn't obvious and as its consequences, there's an extreme contradiction with the term ``extreme R-SuperHyperMatching'' since the maximum extreme cardinality never happens for this extreme style of the extreme SuperHyperSet and beyond that there's no extreme connection inside as mentioned in first extreme case in the forms of drawback for this selected extreme SuperHyperSet. Let
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Comes up. This extreme case implies having the extreme style of on-quasi-triangle extreme style on the every extreme elements of this extreme SuperHyperSet. Precisely, the extreme R-SuperHyperMatching is the extreme SuperHyperSet of the extreme SuperHyperVertices such that some extreme amount of the extreme SuperHyperVertices are on-quasi-triangle extreme style. The extreme cardinality of the v SuperHypeSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Is the maximum in comparison to the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But the lower extreme bound is up. Thus the minimum extreme cardinality of the maximum extreme cardinality ends up the extreme discussion. The first extreme term refers to the extreme setting of the extreme SuperHyperGraph but this key point is enough since there's an extreme SuperHyperClass of an extreme SuperHyperGraph has no on-quasi-triangle extreme style amid some amount of its extreme SuperHyperVertices. This extreme setting of the extreme SuperHyperModel proposes an extreme SuperHyperSet has only some amount  extreme SuperHyperVertices from one extreme SuperHyperEdge such that there's no extreme amount of extreme SuperHyperEdges more than one involving these some amount of these extreme SuperHyperVertices. The extreme cardinality of this extreme SuperHyperSet is the maximum and the extreme case is occurred in the minimum extreme situation. To sum them up, the extreme SuperHyperSet
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Has the maximum extreme cardinality such that
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Contains some extreme SuperHyperVertices such that there's distinct-covers-order-amount extreme SuperHyperEdges for amount of extreme SuperHyperVertices taken from the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 It means that the extreme SuperHyperSet of the extreme SuperHyperVertices
  $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an extreme  R-SuperHyperMatching for the extreme SuperHyperGraph as used extreme background in the extreme terms of worst extreme case and the common theme of the lower extreme bound occurred in the specific extreme SuperHyperClasses of the extreme SuperHyperGraphs which are extreme free-quasi-triangle.
  \\
Assume an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$  extreme number of the extreme SuperHyperVertices. Then every extreme SuperHyperVertex has at least no extreme SuperHyperEdge with others in common. Thus those extreme SuperHyperVertices have the eligibles to be contained in an extreme R-SuperHyperMatching. Those extreme SuperHyperVertices are potentially included in an extreme  style-R-SuperHyperMatching. Formally, consider
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
Are the extreme SuperHyperVertices of an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z.$$ where the $\sim$ isn't an equivalence relation but only the symmetric relation on the extreme  SuperHyperVertices of the extreme SuperHyperGraph. The formal definition is as follows. $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z$$ if and only if $Z_i$ and $Z_j$ are the extreme SuperHyperVertices and there's only and only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ between the extreme SuperHyperVertices $Z_i$ and $Z_j.$ The other definition for the extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ in the terms of extreme R-SuperHyperMatching is
$$\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}.$$
This definition coincides with the definition of the extreme R-SuperHyperMatching but with slightly differences in the maximum extreme cardinality amid those extreme type-SuperHyperSets of the extreme SuperHyperVertices. Thus the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$\max_z|\{Z_1,Z_2,\ldots,Z_z~|~Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}|_{\text{extreme cardinality}},$$
and
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
is formalized with mathematical literatures on the extreme R-SuperHyperMatching. Let $Z_i\overset E\sim Z_j,$ be defined as $Z_i$ and $Z_j$ are the extreme SuperHyperVertices belong to the extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus,  
$$E=\{Z_1,Z_2,\ldots,Z_z~|~Z_i\overset E\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}.$$
Or
$$\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But with the slightly differences,
\begin{eqnarray*}
 &&
 \text{extreme R-SuperHyperMatching}=
 \\&&
 \{Z_1,Z_2,\ldots,Z_z~|~\forall i\neq j,~ i,j=1,2,\ldots,z,~\exists E_x,~Z_i\overset{E_x}\sim Z_j,\}.
\end{eqnarray*}
\begin{eqnarray*}
 &&
 \text{extreme R-SuperHyperMatching}=
 \\&&
V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.
\end{eqnarray*}
Thus $E\in E_{ESHG:(V,E)}$ is an extreme quasi-R-SuperHyperMatching where $E\in E_{ESHG:(V,E)}$ is fixed that means $E_x=E\in E_{ESHG:(V,E)}.$ for all extreme intended SuperHyperVertices but in an extreme SuperHyperMatching, $E_x=E\in E_{ESHG:(V,E)}$ could be different and it's not unique. To sum them up, in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ If an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$ extreme SuperHyperVertices, then the extreme cardinality of the extreme R-SuperHyperMatching is at least
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
 It's straightforward that  the extreme cardinality of the extreme R-SuperHyperMatching is at least the maximum extreme number of extreme SuperHyperVertices of the extreme SuperHyperEdges with the maximum number of the extreme SuperHyperEdges. In other words, the maximum number of the extreme SuperHyperEdges contains the maximum extreme number of extreme SuperHyperVertices are renamed to extreme SuperHyperMatching in some cases but the maximum number of the  extreme SuperHyperEdge with the maximum extreme number of extreme SuperHyperVertices, has the extreme SuperHyperVertices are contained in an extreme R-SuperHyperMatching.
\\
The obvious SuperHyperGraph has no extreme SuperHyperEdges. But the non-obvious extreme SuperHyperModel is up. The quasi-SuperHyperModel addresses some issues about the extreme optimal SuperHyperObject. It specially delivers some remarks on the extreme SuperHyperSet of the extreme SuperHyperVertices such that there's distinct amount of extreme SuperHyperEdges for distinct amount of extreme SuperHyperVertices up to all  taken from that extreme SuperHyperSet of the extreme SuperHyperVertices but this extreme SuperHyperSet of the extreme SuperHyperVertices is either has the maximum extreme SuperHyperCardinality or it doesn't have  maximum extreme SuperHyperCardinality. In a non-obvious SuperHyperModel, there's at least one extreme SuperHyperEdge containing at least all extreme SuperHyperVertices. Thus it forms an extreme quasi-R-SuperHyperMatching where the extreme completion of the extreme incidence is up in that.  Thus it's, literarily, an extreme embedded R-SuperHyperMatching. The SuperHyperNotions of embedded SuperHyperSet and quasi-SuperHyperSet coincide. In the original setting, these types of SuperHyperSets only don't satisfy on the maximum SuperHyperCardinality. Thus the embedded setting is elected such that those SuperHyperSets have the maximum extreme SuperHyperCardinality and they're extreme SuperHyperOptimal. The less than two distinct types of extreme SuperHyperVertices are included in the minimum extreme style of the embedded extreme R-SuperHyperMatching. The interior types of the extreme SuperHyperVertices are deciders. Since the extreme number of SuperHyperNeighbors are only  affected by the interior extreme SuperHyperVertices. The common connections, more precise and more formal, the perfect unique connections inside the extreme SuperHyperSet for any distinct types of extreme SuperHyperVertices pose the extreme R-SuperHyperMatching. Thus extreme exterior SuperHyperVertices could be used only in one extreme SuperHyperEdge and in extreme SuperHyperRelation with the interior extreme SuperHyperVertices in that  extreme SuperHyperEdge. In the embedded extreme SuperHyperMatching, there's the usage of exterior extreme SuperHyperVertices since they've more connections inside more than outside. Thus the title ``exterior'' is more relevant than the title ``interior''. One extreme SuperHyperVertex has no connection, inside. Thus, the extreme SuperHyperSet of the extreme SuperHyperVertices with one SuperHyperElement has been ignored in the exploring to lead on the optimal case implying the extreme R-SuperHyperMatching. The extreme R-SuperHyperMatching with the exclusion of the exclusion of all  extreme SuperHyperVertices in one extreme SuperHyperEdge and with other terms, the extreme R-SuperHyperMatching with the inclusion of all extreme SuperHyperVertices in one extreme SuperHyperEdge, is an extreme quasi-R-SuperHyperMatching. To sum them up, in a connected non-obvious extreme  SuperHyperGraph $ESHG:(V,E).$ There's only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only the maximum possibilities of the distinct interior extreme SuperHyperVertices inside of any given extreme quasi-R-SuperHyperMatching minus all  extreme SuperHypeNeighbor to some of them but not all of them. In other words, there's only an unique extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only two distinct extreme SuperHyperVertices in an  extreme quasi-R-SuperHyperMatching, minus all  extreme SuperHypeNeighbor to some of them but not all of them.
\\
The main definition of the extreme R-SuperHyperMatching has two titles. An extreme quasi-R-SuperHyperMatching and its corresponded quasi-maximum extreme R-SuperHyperCardinality are two titles in the terms of quasi-R-styles. For any extreme number, there's an extreme quasi-R-SuperHyperMatching with that quasi-maximum extreme SuperHyperCardinality in the terms of the embedded extreme SuperHyperGraph. If there's an embedded extreme SuperHyperGraph, then the extreme quasi-SuperHyperNotions lead us to take the collection of all the extreme quasi-R-SuperHyperMatchings for all extreme numbers less than its extreme corresponded maximum number. The essence of the extreme SuperHyperMatching ends up but this essence starts up in the terms of the extreme quasi-R-SuperHyperMatching, again and more in the operations of collecting all the extreme quasi-R-SuperHyperMatchings acted on the all possible used formations of the extreme SuperHyperGraph to achieve one extreme number. This extreme number is considered as the equivalence class for all corresponded quasi-R-SuperHyperMatchings. Let $z_{\text{Extreme Number}},S_{\text{Extreme SuperHyperSet}}$ and $G_{\text{Extreme SuperHyperMatching}}$ be an extreme number, an extreme SuperHyperSet and an extreme SuperHyperMatching. Then
\begin{eqnarray*}
&&[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}\}.
\end{eqnarray*}
As its consequences, the formal definition of the extreme SuperHyperMatching is re-formalized and redefined as follows.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}\}.
\end{eqnarray*}
To get more precise perceptions, the follow-up expressions propose another formal technical definition for the extreme SuperHyperMatching.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
In more concise and more convenient ways, the modified definition for the extreme SuperHyperMatching poses the upcoming expressions.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
To translate the statement to this mathematical literature, the  formulae will be revised.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
And then,
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
To get more visions in the closer look-up, there's an overall overlook.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Now, the extension of these types of approaches is up. Since the new term, ``extreme SuperHyperNeighborhood'', could be redefined as the collection of the extreme SuperHyperVertices such that any amount of its extreme SuperHyperVertices are incident to an extreme  SuperHyperEdge. It's, literarily,  another name for ``extreme  Quasi-SuperHyperMatching'' but, precisely, it's the generalization of  ``extreme  Quasi-SuperHyperMatching'' since ``extreme Quasi-SuperHyperMatching'' happens ``extreme SuperHyperMatching'' in an extreme SuperHyperGraph as initial framework and background but ``extreme SuperHyperNeighborhood'' may not happens ``extreme SuperHyperMatching'' in an extreme SuperHyperGraph as initial framework and preliminarily background since there are some ambiguities about the extreme SuperHyperCardinality arise from it. To get orderly keywords, the terms, ``extreme SuperHyperNeighborhood'',  ``extreme Quasi-SuperHyperMatching'', and  ``extreme SuperHyperMatching'' are up.
\\
Thus, let $z_{\text{Extreme Number}},N_{\text{Extreme SuperHyperNeighborhood}}$ and $G_{\text{Extreme SuperHyperMatching}}$ be an extreme number, an extreme SuperHyperNeighborhood and an extreme SuperHyperMatching and the new terms are up.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
And with go back to initial structure,
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Thus, in a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\\
  To make sense with the precise words in the terms of ``R-', the follow-up illustrations are coming up.
  \\
  The following extreme SuperHyperSet  of extreme  SuperHyperVertices is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching.
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   The extreme SuperHyperSet of extreme SuperHyperVertices,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperVertices,  
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an \underline{\textbf{extreme R-SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge amid some extreme SuperHyperVertices instead of all given by \underline{\textbf{extreme SuperHyperMatching}} is related to the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   There's   \underline{not} only \underline{\textbf{one}} extreme SuperHyperVertex \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{one}} extreme SuperHyperVertex. But the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   doesn't have less than two  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet since they've come from at least so far an SuperHyperEdge. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperVertices,
 $$(V\setminus V\setminus \{x,z\})\cup\{xy\}$$ or $$(V\setminus V\setminus \{x,z\})\cup\{zy\}$$
   is an  extreme R-SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices instead of all given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme SuperHyperEdge for some amount extreme  SuperHyperVertices instead of all given by that extreme type-SuperHyperSet called the  extreme SuperHyperMatching. There isn't  only less than two extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   Thus the non-obvious  extreme R-SuperHyperMatching,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  is up. The non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is  the extreme SuperHyperSet, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 does includes only less than two SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E)$ but it's impossible in the case, they've corresponded to an SuperHyperEdge. It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  R-SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme R-SuperHyperMatching}},
\end{center}
 is only and only
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$  with a illustrated SuperHyperModeling. It's also, not only an extreme free-triangle embedded SuperHyperModel and an extreme on-triangle embedded SuperHyperModel but also it's an extreme stable embedded SuperHyperModel. But all only non-obvious simple extreme type-SuperHyperSets of the extreme  R-SuperHyperMatching amid those obvious simple extreme type-SuperHyperSets of the extreme SuperHyperMatching, are
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
\\
To sum them up,  assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 is an extreme R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of an extreme  R-SuperHyperMatching is the cardinality of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  \\
To sum them up,  in a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\\
Assume a connected extreme  SuperHyperGraph $ESHG:(V,E).$ Let an extreme SuperHyperEdge $ESHE: E\in E_{ESHG:(V,E)}$ has some extreme SuperHyperVertices $r.$ Consider all extreme numbers of those extreme SuperHyperVertices from that extreme SuperHyperEdge excluding excluding more than $r$ distinct extreme SuperHyperVertices, exclude to any given extreme SuperHyperSet of the extreme SuperHyperVertices. Consider there's an extreme  R-SuperHyperMatching with the least cardinality, the lower sharp extreme bound for extreme cardinality. Assume a connected extreme SuperHyperGraph $ESHG:(V,E).$ The extreme SuperHyperSet of the extreme SuperHyperVertices $ V_{ESHE}\setminus \{z\}$ is an extreme SuperHyperSet $S$ of  the extreme SuperHyperVertices such that there's an extreme SuperHyperEdge to have  some extreme SuperHyperVertices uniquely but it isn't an extreme R-SuperHyperMatching. Since it doesn't have  \underline{\textbf{the maximum extreme cardinality}} of an extreme SuperHyperSet $S$ of  extreme SuperHyperVertices such that there's an extreme SuperHyperEdge to have some SuperHyperVertices uniquely. The extreme SuperHyperSet of the extreme SuperHyperVertices $V_{ESHE}\cup \{z\}$ is the maximum extreme cardinality of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices but it isn't an extreme R-SuperHyperMatching. Since it \textbf{\underline{doesn't do}} the extreme procedure such that such that there's an extreme SuperHyperEdge to have some extreme  SuperHyperVertices uniquely  [there are at least one extreme SuperHyperVertex outside  implying there's, sometimes in  the connected extreme SuperHyperGraph $ESHG:(V,E),$ an extreme SuperHyperVertex, titled its extreme SuperHyperNeighbor,  to that extreme SuperHyperVertex in the extreme SuperHyperSet $S$ so as $S$ doesn't do ``the extreme procedure''.]. There's  only \textbf{\underline{one}} extreme SuperHyperVertex    \textbf{\underline{outside}} the intended extreme SuperHyperSet, $V_{ESHE}\cup \{z\},$ in the terms of extreme SuperHyperNeighborhood. Thus the obvious extreme R-SuperHyperMatching,  $V_{ESHE}$ is up. The obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching,  $V_{ESHE},$ \textbf{\underline{is}} an extreme SuperHyperSet, $V_{ESHE},$  \textbf{\underline{includes}} only \textbf{\underline{all}}  extreme SuperHyperVertices does forms any kind of extreme pairs are titled   \underline{extreme SuperHyperNeighbors} in a connected extreme SuperHyperGraph $ESHG:(V,E).$ Since the extreme SuperHyperSet of the extreme SuperHyperVertices $V_{ESHE},$ is the \textbf{\underline{maximum extreme SuperHyperCardinality}} of an extreme SuperHyperSet $S$ of  extreme SuperHyperVertices  \textbf{\underline{such that}}  there's an extreme SuperHyperEdge to have some extreme SuperHyperVertices uniquely. Thus, in a connected extreme SuperHyperGraph $ESHG:(V,E).$ Any extreme R-SuperHyperMatching only contains all interior extreme SuperHyperVertices and all exterior extreme SuperHyperVertices from the unique extreme SuperHyperEdge where there's any of them has all possible  extreme SuperHyperNeighbors in and there's all  extreme SuperHyperNeighborhoods in with no exception minus all  extreme SuperHypeNeighbors to some of them not all of them  but everything is possible about extreme SuperHyperNeighborhoods and extreme SuperHyperNeighbors out.
\\
The SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
 \end{proof}

 \begin{example}\label{118EXM19a}
In the Figure \eqref{118NSHG19a}, the connected extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained extreme SuperHyperSet, in the extreme SuperHyperModel \eqref{118NSHG19a},
 is the extreme  SuperHyperMatching.
\begin{figure}
\includegraphics[width=100mm]{118NSHG19.png}
 \caption{An extreme SuperHyperCycle Associated to the extreme Notions of  extreme SuperHyperMatching in the extreme Example \eqref{118EXM19a}}
 \label{118NSHG19a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected   extreme SuperHyperStar $ESHS:(V,E).$ Then an extreme quasi-R-SuperHyperMatching is an extreme SuperHyperSet of the interior extreme SuperHyperVertices, corresponded to an extreme SuperHyperEdge. An extreme  quasi-R-SuperHyperMatching has the extreme number of the extreme cardinality of the one extreme SuperHyperEdge.  Also, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E\in E_{ESHG:(V,E)}\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=\sum_{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}z^{|E|_{\text{Extreme Cardinality}}~|~E:\in E_{ESHG:(V,E)}}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t},\ldots.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=z^{s}+z^{t}+,\ldots.
\end{eqnarray*}
 \end{proposition}
\begin{proof}
Assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ The SuperHyperSet of the SuperHyperVertices $V\setminus V\setminus \{z\}$ isn't a quasi-R-SuperHyperMatching since neither amount of extreme SuperHyperEdges nor amount of SuperHyperVertices where amount refers to the extreme number of SuperHyperVertices(-/SuperHyperEdges) more than one to form any kind of SuperHyperEdges or any number of SuperHyperEdges. Let us consider the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
This extreme SuperHyperSet of the extreme SuperHyperVertices has the eligibilities to propose property such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices but the maximum extreme cardinality indicates that these extreme  type-SuperHyperSets couldn't give us the extreme lower bound in the term of extreme sharpness. In other words, the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
of the extreme SuperHyperVertices implies at least on-quasi-triangle style is up but sometimes the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
 of the extreme SuperHyperVertices is free-quasi-triangle and it doesn't make a contradiction to the supposition on the connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Thus the minimum case never happens in the generality of the connected loopless neutrosophic SuperHyperGraphs. Thus if we assume in the worst case, literally,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is a quasi-R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of a  quasi-R-SuperHyperMatching is the cardinality of
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 Then we've lost some connected loopless neutrosophic SuperHyperClasses of the connected loopless neutrosophic SuperHyperGraphs titled free-triangle, on-triangle, and their quasi-types but the SuperHyperStable is only up in this quasi-R-SuperHyperMatching. It's the contradiction to that fact on the generality. There are some counterexamples to deny this statement. One of them comes from the setting of the graph titled path and cycle as the counterexamples-classes or reversely direction star as the examples-classes, are well-known classes in that setting and they could be considered as the examples-classes and counterexamples-classes for the tight bound of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  Let $V\setminus V\setminus \{z\}$ in mind. There's no necessity on the SuperHyperEdge since we need at least two SuperHyperVertices to form a SuperHyperEdge. It doesn't withdraw the principles of the main definition since there's no condition to be satisfied but the condition is on the existence of the SuperHyperEdge instead of acting on the SuperHyperVertices. In other words, if there's a SuperHyperEdge, then the extreme SuperHyperSet has the necessary condition for the intended definition to be applied. Thus the $V\setminus V\setminus \{z\}$ is withdrawn not by the conditions of the main definition but by the necessity of the pre-condition on the usage of the main definition.
  \\
  The extreme structure of the extreme R-SuperHyperMatching decorates the extreme SuperHyperVertices don't have received any extreme connections so as this extreme style implies different versions of extreme SuperHyperEdges with the maximum extreme cardinality in the terms of extreme SuperHyperVertices are spotlight. The lower extreme bound is to have the maximum extreme groups of extreme SuperHyperVertices have perfect extreme connections inside each of SuperHyperEdges and the outside of this extreme SuperHyperSet doesn't matter but regarding the connectedness of the used extreme SuperHyperGraph arising from its extreme properties taken from the fact that it's simple. If there's no more than one extreme SuperHyperVertex in the targeted extreme SuperHyperSet, then there's no extreme connection. Furthermore, the extreme existence of one extreme SuperHyperVertex has no  extreme effect to talk about the extreme R-SuperHyperMatching. Since at least two extreme SuperHyperVertices involve to make a title in the extreme background of the extreme SuperHyperGraph. The extreme SuperHyperGraph is obvious if it has no extreme SuperHyperEdge but at least two extreme SuperHyperVertices make the extreme version of extreme SuperHyperEdge. Thus in the extreme setting of non-obvious extreme SuperHyperGraph, there are at least one extreme SuperHyperEdge. It's necessary to mention that the word ``Simple'' is used as extreme adjective for the initial extreme SuperHyperGraph, induces there's no extreme  appearance of the loop extreme version of the extreme SuperHyperEdge and this extreme SuperHyperGraph is said to be loopless. The extreme adjective ``loop'' on the basic extreme framework engages one extreme SuperHyperVertex but it never happens in this extreme setting. With these extreme bases, on an extreme SuperHyperGraph, there's at least one extreme SuperHyperEdge thus there's at least an extreme R-SuperHyperMatching has the extreme cardinality of an extreme SuperHyperEdge. Thus, an extreme R-SuperHyperMatching has the extreme cardinality at least an extreme SuperHyperEdge. Assume an extreme SuperHyperSet $V\setminus V\setminus \{z\}.$ This extreme SuperHyperSet isn't an extreme R-SuperHyperMatching since either the extreme SuperHyperGraph is an obvious extreme SuperHyperModel thus it never happens since there's no extreme usage of this extreme framework and even more there's no extreme connection inside or the extreme SuperHyperGraph isn't obvious and as its consequences, there's an extreme contradiction with the term ``extreme R-SuperHyperMatching'' since the maximum extreme cardinality never happens for this extreme style of the extreme SuperHyperSet and beyond that there's no extreme connection inside as mentioned in first extreme case in the forms of drawback for this selected extreme SuperHyperSet. Let
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Comes up. This extreme case implies having the extreme style of on-quasi-triangle extreme style on the every extreme elements of this extreme SuperHyperSet. Precisely, the extreme R-SuperHyperMatching is the extreme SuperHyperSet of the extreme SuperHyperVertices such that some extreme amount of the extreme SuperHyperVertices are on-quasi-triangle extreme style. The extreme cardinality of the v SuperHypeSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Is the maximum in comparison to the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But the lower extreme bound is up. Thus the minimum extreme cardinality of the maximum extreme cardinality ends up the extreme discussion. The first extreme term refers to the extreme setting of the extreme SuperHyperGraph but this key point is enough since there's an extreme SuperHyperClass of an extreme SuperHyperGraph has no on-quasi-triangle extreme style amid some amount of its extreme SuperHyperVertices. This extreme setting of the extreme SuperHyperModel proposes an extreme SuperHyperSet has only some amount  extreme SuperHyperVertices from one extreme SuperHyperEdge such that there's no extreme amount of extreme SuperHyperEdges more than one involving these some amount of these extreme SuperHyperVertices. The extreme cardinality of this extreme SuperHyperSet is the maximum and the extreme case is occurred in the minimum extreme situation. To sum them up, the extreme SuperHyperSet
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Has the maximum extreme cardinality such that
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Contains some extreme SuperHyperVertices such that there's distinct-covers-order-amount extreme SuperHyperEdges for amount of extreme SuperHyperVertices taken from the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 It means that the extreme SuperHyperSet of the extreme SuperHyperVertices
  $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an extreme  R-SuperHyperMatching for the extreme SuperHyperGraph as used extreme background in the extreme terms of worst extreme case and the common theme of the lower extreme bound occurred in the specific extreme SuperHyperClasses of the extreme SuperHyperGraphs which are extreme free-quasi-triangle.
  \\
Assume an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$  extreme number of the extreme SuperHyperVertices. Then every extreme SuperHyperVertex has at least no extreme SuperHyperEdge with others in common. Thus those extreme SuperHyperVertices have the eligibles to be contained in an extreme R-SuperHyperMatching. Those extreme SuperHyperVertices are potentially included in an extreme  style-R-SuperHyperMatching. Formally, consider
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
Are the extreme SuperHyperVertices of an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z.$$ where the $\sim$ isn't an equivalence relation but only the symmetric relation on the extreme  SuperHyperVertices of the extreme SuperHyperGraph. The formal definition is as follows. $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z$$ if and only if $Z_i$ and $Z_j$ are the extreme SuperHyperVertices and there's only and only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ between the extreme SuperHyperVertices $Z_i$ and $Z_j.$ The other definition for the extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ in the terms of extreme R-SuperHyperMatching is
$$\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}.$$
This definition coincides with the definition of the extreme R-SuperHyperMatching but with slightly differences in the maximum extreme cardinality amid those extreme type-SuperHyperSets of the extreme SuperHyperVertices. Thus the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$\max_z|\{Z_1,Z_2,\ldots,Z_z~|~Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}|_{\text{extreme cardinality}},$$
and
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
is formalized with mathematical literatures on the extreme R-SuperHyperMatching. Let $Z_i\overset E\sim Z_j,$ be defined as $Z_i$ and $Z_j$ are the extreme SuperHyperVertices belong to the extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus,  
$$E=\{Z_1,Z_2,\ldots,Z_z~|~Z_i\overset E\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}.$$
Or
$$\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But with the slightly differences,
\begin{eqnarray*}
 &&
 \text{extreme R-SuperHyperMatching}=
 \\&&
 \{Z_1,Z_2,\ldots,Z_z~|~\forall i\neq j,~ i,j=1,2,\ldots,z,~\exists E_x,~Z_i\overset{E_x}\sim Z_j,\}.
\end{eqnarray*}
\begin{eqnarray*}
 &&
 \text{extreme R-SuperHyperMatching}=
 \\&&
V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.
\end{eqnarray*}
Thus $E\in E_{ESHG:(V,E)}$ is an extreme quasi-R-SuperHyperMatching where $E\in E_{ESHG:(V,E)}$ is fixed that means $E_x=E\in E_{ESHG:(V,E)}.$ for all extreme intended SuperHyperVertices but in an extreme SuperHyperMatching, $E_x=E\in E_{ESHG:(V,E)}$ could be different and it's not unique. To sum them up, in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ If an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$ extreme SuperHyperVertices, then the extreme cardinality of the extreme R-SuperHyperMatching is at least
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
 It's straightforward that  the extreme cardinality of the extreme R-SuperHyperMatching is at least the maximum extreme number of extreme SuperHyperVertices of the extreme SuperHyperEdges with the maximum number of the extreme SuperHyperEdges. In other words, the maximum number of the extreme SuperHyperEdges contains the maximum extreme number of extreme SuperHyperVertices are renamed to extreme SuperHyperMatching in some cases but the maximum number of the  extreme SuperHyperEdge with the maximum extreme number of extreme SuperHyperVertices, has the extreme SuperHyperVertices are contained in an extreme R-SuperHyperMatching.
\\
The obvious SuperHyperGraph has no extreme SuperHyperEdges. But the non-obvious extreme SuperHyperModel is up. The quasi-SuperHyperModel addresses some issues about the extreme optimal SuperHyperObject. It specially delivers some remarks on the extreme SuperHyperSet of the extreme SuperHyperVertices such that there's distinct amount of extreme SuperHyperEdges for distinct amount of extreme SuperHyperVertices up to all  taken from that extreme SuperHyperSet of the extreme SuperHyperVertices but this extreme SuperHyperSet of the extreme SuperHyperVertices is either has the maximum extreme SuperHyperCardinality or it doesn't have  maximum extreme SuperHyperCardinality. In a non-obvious SuperHyperModel, there's at least one extreme SuperHyperEdge containing at least all extreme SuperHyperVertices. Thus it forms an extreme quasi-R-SuperHyperMatching where the extreme completion of the extreme incidence is up in that.  Thus it's, literarily, an extreme embedded R-SuperHyperMatching. The SuperHyperNotions of embedded SuperHyperSet and quasi-SuperHyperSet coincide. In the original setting, these types of SuperHyperSets only don't satisfy on the maximum SuperHyperCardinality. Thus the embedded setting is elected such that those SuperHyperSets have the maximum extreme SuperHyperCardinality and they're extreme SuperHyperOptimal. The less than two distinct types of extreme SuperHyperVertices are included in the minimum extreme style of the embedded extreme R-SuperHyperMatching. The interior types of the extreme SuperHyperVertices are deciders. Since the extreme number of SuperHyperNeighbors are only  affected by the interior extreme SuperHyperVertices. The common connections, more precise and more formal, the perfect unique connections inside the extreme SuperHyperSet for any distinct types of extreme SuperHyperVertices pose the extreme R-SuperHyperMatching. Thus extreme exterior SuperHyperVertices could be used only in one extreme SuperHyperEdge and in extreme SuperHyperRelation with the interior extreme SuperHyperVertices in that  extreme SuperHyperEdge. In the embedded extreme SuperHyperMatching, there's the usage of exterior extreme SuperHyperVertices since they've more connections inside more than outside. Thus the title ``exterior'' is more relevant than the title ``interior''. One extreme SuperHyperVertex has no connection, inside. Thus, the extreme SuperHyperSet of the extreme SuperHyperVertices with one SuperHyperElement has been ignored in the exploring to lead on the optimal case implying the extreme R-SuperHyperMatching. The extreme R-SuperHyperMatching with the exclusion of the exclusion of all  extreme SuperHyperVertices in one extreme SuperHyperEdge and with other terms, the extreme R-SuperHyperMatching with the inclusion of all extreme SuperHyperVertices in one extreme SuperHyperEdge, is an extreme quasi-R-SuperHyperMatching. To sum them up, in a connected non-obvious extreme  SuperHyperGraph $ESHG:(V,E).$ There's only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only the maximum possibilities of the distinct interior extreme SuperHyperVertices inside of any given extreme quasi-R-SuperHyperMatching minus all  extreme SuperHypeNeighbor to some of them but not all of them. In other words, there's only an unique extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only two distinct extreme SuperHyperVertices in an  extreme quasi-R-SuperHyperMatching, minus all  extreme SuperHypeNeighbor to some of them but not all of them.
\\
The main definition of the extreme R-SuperHyperMatching has two titles. An extreme quasi-R-SuperHyperMatching and its corresponded quasi-maximum extreme R-SuperHyperCardinality are two titles in the terms of quasi-R-styles. For any extreme number, there's an extreme quasi-R-SuperHyperMatching with that quasi-maximum extreme SuperHyperCardinality in the terms of the embedded extreme SuperHyperGraph. If there's an embedded extreme SuperHyperGraph, then the extreme quasi-SuperHyperNotions lead us to take the collection of all the extreme quasi-R-SuperHyperMatchings for all extreme numbers less than its extreme corresponded maximum number. The essence of the extreme SuperHyperMatching ends up but this essence starts up in the terms of the extreme quasi-R-SuperHyperMatching, again and more in the operations of collecting all the extreme quasi-R-SuperHyperMatchings acted on the all possible used formations of the extreme SuperHyperGraph to achieve one extreme number. This extreme number is considered as the equivalence class for all corresponded quasi-R-SuperHyperMatchings. Let $z_{\text{Extreme Number}},S_{\text{Extreme SuperHyperSet}}$ and $G_{\text{Extreme SuperHyperMatching}}$ be an extreme number, an extreme SuperHyperSet and an extreme SuperHyperMatching. Then
\begin{eqnarray*}
&&[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}\}.
\end{eqnarray*}
As its consequences, the formal definition of the extreme SuperHyperMatching is re-formalized and redefined as follows.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}\}.
\end{eqnarray*}
To get more precise perceptions, the follow-up expressions propose another formal technical definition for the extreme SuperHyperMatching.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
In more concise and more convenient ways, the modified definition for the extreme SuperHyperMatching poses the upcoming expressions.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
To translate the statement to this mathematical literature, the  formulae will be revised.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
And then,
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
To get more visions in the closer look-up, there's an overall overlook.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Now, the extension of these types of approaches is up. Since the new term, ``extreme SuperHyperNeighborhood'', could be redefined as the collection of the extreme SuperHyperVertices such that any amount of its extreme SuperHyperVertices are incident to an extreme  SuperHyperEdge. It's, literarily,  another name for ``extreme  Quasi-SuperHyperMatching'' but, precisely, it's the generalization of  ``extreme  Quasi-SuperHyperMatching'' since ``extreme Quasi-SuperHyperMatching'' happens ``extreme SuperHyperMatching'' in an extreme SuperHyperGraph as initial framework and background but ``extreme SuperHyperNeighborhood'' may not happens ``extreme SuperHyperMatching'' in an extreme SuperHyperGraph as initial framework and preliminarily background since there are some ambiguities about the extreme SuperHyperCardinality arise from it. To get orderly keywords, the terms, ``extreme SuperHyperNeighborhood'',  ``extreme Quasi-SuperHyperMatching'', and  ``extreme SuperHyperMatching'' are up.
\\
Thus, let $z_{\text{Extreme Number}},N_{\text{Extreme SuperHyperNeighborhood}}$ and $G_{\text{Extreme SuperHyperMatching}}$ be an extreme number, an extreme SuperHyperNeighborhood and an extreme SuperHyperMatching and the new terms are up.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
And with go back to initial structure,
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Thus, in a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\\
  To make sense with the precise words in the terms of ``R-', the follow-up illustrations are coming up.
  \\
  The following extreme SuperHyperSet  of extreme  SuperHyperVertices is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching.
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   The extreme SuperHyperSet of extreme SuperHyperVertices,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperVertices,  
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an \underline{\textbf{extreme R-SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge amid some extreme SuperHyperVertices instead of all given by \underline{\textbf{extreme SuperHyperMatching}} is related to the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   There's   \underline{not} only \underline{\textbf{one}} extreme SuperHyperVertex \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{one}} extreme SuperHyperVertex. But the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   doesn't have less than two  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet since they've come from at least so far an SuperHyperEdge. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperVertices,
 $$(V\setminus V\setminus \{x,z\})\cup\{xy\}$$ or $$(V\setminus V\setminus \{x,z\})\cup\{zy\}$$
   is an  extreme R-SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices instead of all given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme SuperHyperEdge for some amount extreme  SuperHyperVertices instead of all given by that extreme type-SuperHyperSet called the  extreme SuperHyperMatching. There isn't  only less than two extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   Thus the non-obvious  extreme R-SuperHyperMatching,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  is up. The non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is  the extreme SuperHyperSet, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 does includes only less than two SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E)$ but it's impossible in the case, they've corresponded to an SuperHyperEdge. It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  R-SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme R-SuperHyperMatching}},
\end{center}
 is only and only
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$  with a illustrated SuperHyperModeling. It's also, not only an extreme free-triangle embedded SuperHyperModel and an extreme on-triangle embedded SuperHyperModel but also it's an extreme stable embedded SuperHyperModel. But all only non-obvious simple extreme type-SuperHyperSets of the extreme  R-SuperHyperMatching amid those obvious simple extreme type-SuperHyperSets of the extreme SuperHyperMatching, are
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
\\
To sum them up,  assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 is an extreme R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of an extreme  R-SuperHyperMatching is the cardinality of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  \\
To sum them up,  in a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\\
Assume a connected extreme  SuperHyperGraph $ESHG:(V,E).$ Let an extreme SuperHyperEdge $ESHE: E\in E_{ESHG:(V,E)}$ has some extreme SuperHyperVertices $r.$ Consider all extreme numbers of those extreme SuperHyperVertices from that extreme SuperHyperEdge excluding excluding more than $r$ distinct extreme SuperHyperVertices, exclude to any given extreme SuperHyperSet of the extreme SuperHyperVertices. Consider there's an extreme  R-SuperHyperMatching with the least cardinality, the lower sharp extreme bound for extreme cardinality. Assume a connected extreme SuperHyperGraph $ESHG:(V,E).$ The extreme SuperHyperSet of the extreme SuperHyperVertices $ V_{ESHE}\setminus \{z\}$ is an extreme SuperHyperSet $S$ of  the extreme SuperHyperVertices such that there's an extreme SuperHyperEdge to have  some extreme SuperHyperVertices uniquely but it isn't an extreme R-SuperHyperMatching. Since it doesn't have  \underline{\textbf{the maximum extreme cardinality}} of an extreme SuperHyperSet $S$ of  extreme SuperHyperVertices such that there's an extreme SuperHyperEdge to have some SuperHyperVertices uniquely. The extreme SuperHyperSet of the extreme SuperHyperVertices $V_{ESHE}\cup \{z\}$ is the maximum extreme cardinality of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices but it isn't an extreme R-SuperHyperMatching. Since it \textbf{\underline{doesn't do}} the extreme procedure such that such that there's an extreme SuperHyperEdge to have some extreme  SuperHyperVertices uniquely  [there are at least one extreme SuperHyperVertex outside  implying there's, sometimes in  the connected extreme SuperHyperGraph $ESHG:(V,E),$ an extreme SuperHyperVertex, titled its extreme SuperHyperNeighbor,  to that extreme SuperHyperVertex in the extreme SuperHyperSet $S$ so as $S$ doesn't do ``the extreme procedure''.]. There's  only \textbf{\underline{one}} extreme SuperHyperVertex    \textbf{\underline{outside}} the intended extreme SuperHyperSet, $V_{ESHE}\cup \{z\},$ in the terms of extreme SuperHyperNeighborhood. Thus the obvious extreme R-SuperHyperMatching,  $V_{ESHE}$ is up. The obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching,  $V_{ESHE},$ \textbf{\underline{is}} an extreme SuperHyperSet, $V_{ESHE},$  \textbf{\underline{includes}} only \textbf{\underline{all}}  extreme SuperHyperVertices does forms any kind of extreme pairs are titled   \underline{extreme SuperHyperNeighbors} in a connected extreme SuperHyperGraph $ESHG:(V,E).$ Since the extreme SuperHyperSet of the extreme SuperHyperVertices $V_{ESHE},$ is the \textbf{\underline{maximum extreme SuperHyperCardinality}} of an extreme SuperHyperSet $S$ of  extreme SuperHyperVertices  \textbf{\underline{such that}}  there's an extreme SuperHyperEdge to have some extreme SuperHyperVertices uniquely. Thus, in a connected extreme SuperHyperGraph $ESHG:(V,E).$ Any extreme R-SuperHyperMatching only contains all interior extreme SuperHyperVertices and all exterior extreme SuperHyperVertices from the unique extreme SuperHyperEdge where there's any of them has all possible  extreme SuperHyperNeighbors in and there's all  extreme SuperHyperNeighborhoods in with no exception minus all  extreme SuperHypeNeighbors to some of them not all of them  but everything is possible about extreme SuperHyperNeighborhoods and extreme SuperHyperNeighbors out.
\\
The SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E\in E_{ESHG:(V,E)}\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=\sum_{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}z^{|E|_{\text{Extreme Cardinality}}~|~E:\in E_{ESHG:(V,E)}}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t},\ldots.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=z^{s}+z^{t}+,\ldots.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E\in E_{ESHG:(V,E)}\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=\sum_{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}z^{|E|_{\text{Extreme Cardinality}}~|~E:\in E_{ESHG:(V,E)}}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t},\ldots.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=z^{s}+z^{t}+,\ldots.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E\in E_{ESHG:(V,E)}\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=\sum_{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}z^{|E|_{\text{Extreme Cardinality}}~|~E:\in E_{ESHG:(V,E)}}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t},\ldots.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=z^{s}+z^{t}+,\ldots.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E\in E_{ESHG:(V,E)}\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=\sum_{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}z^{|E|_{\text{Extreme Cardinality}}~|~E:\in E_{ESHG:(V,E)}}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t},\ldots.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=z^{s}+z^{t}+,\ldots.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E\in E_{ESHG:(V,E)}\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=\sum_{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}z^{|E|_{\text{Extreme Cardinality}}~|~E:\in E_{ESHG:(V,E)}}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t},\ldots.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=z^{s}+z^{t}+,\ldots.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E\in E_{ESHG:(V,E)}\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=\sum_{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}z^{|E|_{\text{Extreme Cardinality}}~|~E:\in E_{ESHG:(V,E)}}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t},\ldots.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=z^{s}+z^{t}+,\ldots.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E\in E_{ESHG:(V,E)}\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=\sum_{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}z^{|E|_{\text{Extreme Cardinality}}~|~E:\in E_{ESHG:(V,E)}}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t},\ldots.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=z^{s}+z^{t}+,\ldots.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E\in E_{ESHG:(V,E)}\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=\sum_{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}z^{|E|_{\text{Extreme Cardinality}}~|~E:\in E_{ESHG:(V,E)}}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t},\ldots.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=z^{s}+z^{t}+,\ldots.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E\in E_{ESHG:(V,E)}\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=\sum_{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}z^{|E|_{\text{Extreme Cardinality}}~|~E:\in E_{ESHG:(V,E)}}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t},\ldots.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=z^{s}+z^{t}+,\ldots.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E\in E_{ESHG:(V,E)}\}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=\sum_{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}z^{|E|_{\text{Extreme Cardinality}}~|~E:\in E_{ESHG:(V,E)}}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t},\ldots.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=z^{s}+z^{t}+,\ldots.
\end{eqnarray*}
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
 \end{proof}

 \begin{example}\label{118EXM20a}
 In the Figure \eqref{118NSHG20a}, the connected extreme SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained extreme SuperHyperSet, by the Algorithm in previous extreme result, of the  extreme SuperHyperVertices of the connected extreme SuperHyperStar $ESHS:(V,E),$ in the extreme SuperHyperModel \eqref{118NSHG20a}, is the  extreme SuperHyperMatching.
\begin{figure}
\includegraphics[width=100mm]{118NSHG20.png}
 \caption{An extreme SuperHyperStar Associated to the extreme Notions of  extreme SuperHyperMatching in the extreme Example \eqref{118EXM20a}}
 \label{118NSHG20a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected   extreme SuperHyperBipartite $ESHB:(V,E).$ Then an extreme  R-SuperHyperMatching is an extreme SuperHyperSet of the interior extreme SuperHyperVertices with no extreme exceptions in the form of interior extreme SuperHyperVertices titled extreme SuperHyperNeighbors.  An extreme  R-SuperHyperMatching has the extreme maximum number of on extreme cardinality of the minimum SuperHyperPart minus those have common extreme SuperHyperNeighbors and not unique extreme SuperHyperNeighbors. Also,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}  
\end{proposition}
\begin{proof}
Assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ The SuperHyperSet of the SuperHyperVertices $V\setminus V\setminus \{z\}$ isn't a quasi-R-SuperHyperMatching since neither amount of extreme SuperHyperEdges nor amount of SuperHyperVertices where amount refers to the extreme number of SuperHyperVertices(-/SuperHyperEdges) more than one to form any kind of SuperHyperEdges or any number of SuperHyperEdges. Let us consider the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
This extreme SuperHyperSet of the extreme SuperHyperVertices has the eligibilities to propose property such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices but the maximum extreme cardinality indicates that these extreme  type-SuperHyperSets couldn't give us the extreme lower bound in the term of extreme sharpness. In other words, the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
of the extreme SuperHyperVertices implies at least on-quasi-triangle style is up but sometimes the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
 of the extreme SuperHyperVertices is free-quasi-triangle and it doesn't make a contradiction to the supposition on the connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Thus the minimum case never happens in the generality of the connected loopless neutrosophic SuperHyperGraphs. Thus if we assume in the worst case, literally,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is a quasi-R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of a  quasi-R-SuperHyperMatching is the cardinality of
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 Then we've lost some connected loopless neutrosophic SuperHyperClasses of the connected loopless neutrosophic SuperHyperGraphs titled free-triangle, on-triangle, and their quasi-types but the SuperHyperStable is only up in this quasi-R-SuperHyperMatching. It's the contradiction to that fact on the generality. There are some counterexamples to deny this statement. One of them comes from the setting of the graph titled path and cycle as the counterexamples-classes or reversely direction star as the examples-classes, are well-known classes in that setting and they could be considered as the examples-classes and counterexamples-classes for the tight bound of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  Let $V\setminus V\setminus \{z\}$ in mind. There's no necessity on the SuperHyperEdge since we need at least two SuperHyperVertices to form a SuperHyperEdge. It doesn't withdraw the principles of the main definition since there's no condition to be satisfied but the condition is on the existence of the SuperHyperEdge instead of acting on the SuperHyperVertices. In other words, if there's a SuperHyperEdge, then the extreme SuperHyperSet has the necessary condition for the intended definition to be applied. Thus the $V\setminus V\setminus \{z\}$ is withdrawn not by the conditions of the main definition but by the necessity of the pre-condition on the usage of the main definition.
  \\
  The extreme structure of the extreme R-SuperHyperMatching decorates the extreme SuperHyperVertices don't have received any extreme connections so as this extreme style implies different versions of extreme SuperHyperEdges with the maximum extreme cardinality in the terms of extreme SuperHyperVertices are spotlight. The lower extreme bound is to have the maximum extreme groups of extreme SuperHyperVertices have perfect extreme connections inside each of SuperHyperEdges and the outside of this extreme SuperHyperSet doesn't matter but regarding the connectedness of the used extreme SuperHyperGraph arising from its extreme properties taken from the fact that it's simple. If there's no more than one extreme SuperHyperVertex in the targeted extreme SuperHyperSet, then there's no extreme connection. Furthermore, the extreme existence of one extreme SuperHyperVertex has no  extreme effect to talk about the extreme R-SuperHyperMatching. Since at least two extreme SuperHyperVertices involve to make a title in the extreme background of the extreme SuperHyperGraph. The extreme SuperHyperGraph is obvious if it has no extreme SuperHyperEdge but at least two extreme SuperHyperVertices make the extreme version of extreme SuperHyperEdge. Thus in the extreme setting of non-obvious extreme SuperHyperGraph, there are at least one extreme SuperHyperEdge. It's necessary to mention that the word ``Simple'' is used as extreme adjective for the initial extreme SuperHyperGraph, induces there's no extreme  appearance of the loop extreme version of the extreme SuperHyperEdge and this extreme SuperHyperGraph is said to be loopless. The extreme adjective ``loop'' on the basic extreme framework engages one extreme SuperHyperVertex but it never happens in this extreme setting. With these extreme bases, on an extreme SuperHyperGraph, there's at least one extreme SuperHyperEdge thus there's at least an extreme R-SuperHyperMatching has the extreme cardinality of an extreme SuperHyperEdge. Thus, an extreme R-SuperHyperMatching has the extreme cardinality at least an extreme SuperHyperEdge. Assume an extreme SuperHyperSet $V\setminus V\setminus \{z\}.$ This extreme SuperHyperSet isn't an extreme R-SuperHyperMatching since either the extreme SuperHyperGraph is an obvious extreme SuperHyperModel thus it never happens since there's no extreme usage of this extreme framework and even more there's no extreme connection inside or the extreme SuperHyperGraph isn't obvious and as its consequences, there's an extreme contradiction with the term ``extreme R-SuperHyperMatching'' since the maximum extreme cardinality never happens for this extreme style of the extreme SuperHyperSet and beyond that there's no extreme connection inside as mentioned in first extreme case in the forms of drawback for this selected extreme SuperHyperSet. Let
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Comes up. This extreme case implies having the extreme style of on-quasi-triangle extreme style on the every extreme elements of this extreme SuperHyperSet. Precisely, the extreme R-SuperHyperMatching is the extreme SuperHyperSet of the extreme SuperHyperVertices such that some extreme amount of the extreme SuperHyperVertices are on-quasi-triangle extreme style. The extreme cardinality of the v SuperHypeSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Is the maximum in comparison to the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But the lower extreme bound is up. Thus the minimum extreme cardinality of the maximum extreme cardinality ends up the extreme discussion. The first extreme term refers to the extreme setting of the extreme SuperHyperGraph but this key point is enough since there's an extreme SuperHyperClass of an extreme SuperHyperGraph has no on-quasi-triangle extreme style amid some amount of its extreme SuperHyperVertices. This extreme setting of the extreme SuperHyperModel proposes an extreme SuperHyperSet has only some amount  extreme SuperHyperVertices from one extreme SuperHyperEdge such that there's no extreme amount of extreme SuperHyperEdges more than one involving these some amount of these extreme SuperHyperVertices. The extreme cardinality of this extreme SuperHyperSet is the maximum and the extreme case is occurred in the minimum extreme situation. To sum them up, the extreme SuperHyperSet
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Has the maximum extreme cardinality such that
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Contains some extreme SuperHyperVertices such that there's distinct-covers-order-amount extreme SuperHyperEdges for amount of extreme SuperHyperVertices taken from the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 It means that the extreme SuperHyperSet of the extreme SuperHyperVertices
  $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an extreme  R-SuperHyperMatching for the extreme SuperHyperGraph as used extreme background in the extreme terms of worst extreme case and the common theme of the lower extreme bound occurred in the specific extreme SuperHyperClasses of the extreme SuperHyperGraphs which are extreme free-quasi-triangle.
  \\
Assume an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$  extreme number of the extreme SuperHyperVertices. Then every extreme SuperHyperVertex has at least no extreme SuperHyperEdge with others in common. Thus those extreme SuperHyperVertices have the eligibles to be contained in an extreme R-SuperHyperMatching. Those extreme SuperHyperVertices are potentially included in an extreme  style-R-SuperHyperMatching. Formally, consider
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
Are the extreme SuperHyperVertices of an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z.$$ where the $\sim$ isn't an equivalence relation but only the symmetric relation on the extreme  SuperHyperVertices of the extreme SuperHyperGraph. The formal definition is as follows. $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z$$ if and only if $Z_i$ and $Z_j$ are the extreme SuperHyperVertices and there's only and only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ between the extreme SuperHyperVertices $Z_i$ and $Z_j.$ The other definition for the extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ in the terms of extreme R-SuperHyperMatching is
$$\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}.$$
This definition coincides with the definition of the extreme R-SuperHyperMatching but with slightly differences in the maximum extreme cardinality amid those extreme type-SuperHyperSets of the extreme SuperHyperVertices. Thus the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$\max_z|\{Z_1,Z_2,\ldots,Z_z~|~Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}|_{\text{extreme cardinality}},$$
and
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
is formalized with mathematical literatures on the extreme R-SuperHyperMatching. Let $Z_i\overset E\sim Z_j,$ be defined as $Z_i$ and $Z_j$ are the extreme SuperHyperVertices belong to the extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus,  
$$E=\{Z_1,Z_2,\ldots,Z_z~|~Z_i\overset E\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}.$$
Or
$$\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But with the slightly differences,
\begin{eqnarray*}
 &&
 \text{extreme R-SuperHyperMatching}=
 \\&&
 \{Z_1,Z_2,\ldots,Z_z~|~\forall i\neq j,~ i,j=1,2,\ldots,z,~\exists E_x,~Z_i\overset{E_x}\sim Z_j,\}.
\end{eqnarray*}
\begin{eqnarray*}
 &&
 \text{extreme R-SuperHyperMatching}=
 \\&&
V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.
\end{eqnarray*}
Thus $E\in E_{ESHG:(V,E)}$ is an extreme quasi-R-SuperHyperMatching where $E\in E_{ESHG:(V,E)}$ is fixed that means $E_x=E\in E_{ESHG:(V,E)}.$ for all extreme intended SuperHyperVertices but in an extreme SuperHyperMatching, $E_x=E\in E_{ESHG:(V,E)}$ could be different and it's not unique. To sum them up, in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ If an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$ extreme SuperHyperVertices, then the extreme cardinality of the extreme R-SuperHyperMatching is at least
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
 It's straightforward that  the extreme cardinality of the extreme R-SuperHyperMatching is at least the maximum extreme number of extreme SuperHyperVertices of the extreme SuperHyperEdges with the maximum number of the extreme SuperHyperEdges. In other words, the maximum number of the extreme SuperHyperEdges contains the maximum extreme number of extreme SuperHyperVertices are renamed to extreme SuperHyperMatching in some cases but the maximum number of the  extreme SuperHyperEdge with the maximum extreme number of extreme SuperHyperVertices, has the extreme SuperHyperVertices are contained in an extreme R-SuperHyperMatching.
\\
The obvious SuperHyperGraph has no extreme SuperHyperEdges. But the non-obvious extreme SuperHyperModel is up. The quasi-SuperHyperModel addresses some issues about the extreme optimal SuperHyperObject. It specially delivers some remarks on the extreme SuperHyperSet of the extreme SuperHyperVertices such that there's distinct amount of extreme SuperHyperEdges for distinct amount of extreme SuperHyperVertices up to all  taken from that extreme SuperHyperSet of the extreme SuperHyperVertices but this extreme SuperHyperSet of the extreme SuperHyperVertices is either has the maximum extreme SuperHyperCardinality or it doesn't have  maximum extreme SuperHyperCardinality. In a non-obvious SuperHyperModel, there's at least one extreme SuperHyperEdge containing at least all extreme SuperHyperVertices. Thus it forms an extreme quasi-R-SuperHyperMatching where the extreme completion of the extreme incidence is up in that.  Thus it's, literarily, an extreme embedded R-SuperHyperMatching. The SuperHyperNotions of embedded SuperHyperSet and quasi-SuperHyperSet coincide. In the original setting, these types of SuperHyperSets only don't satisfy on the maximum SuperHyperCardinality. Thus the embedded setting is elected such that those SuperHyperSets have the maximum extreme SuperHyperCardinality and they're extreme SuperHyperOptimal. The less than two distinct types of extreme SuperHyperVertices are included in the minimum extreme style of the embedded extreme R-SuperHyperMatching. The interior types of the extreme SuperHyperVertices are deciders. Since the extreme number of SuperHyperNeighbors are only  affected by the interior extreme SuperHyperVertices. The common connections, more precise and more formal, the perfect unique connections inside the extreme SuperHyperSet for any distinct types of extreme SuperHyperVertices pose the extreme R-SuperHyperMatching. Thus extreme exterior SuperHyperVertices could be used only in one extreme SuperHyperEdge and in extreme SuperHyperRelation with the interior extreme SuperHyperVertices in that  extreme SuperHyperEdge. In the embedded extreme SuperHyperMatching, there's the usage of exterior extreme SuperHyperVertices since they've more connections inside more than outside. Thus the title ``exterior'' is more relevant than the title ``interior''. One extreme SuperHyperVertex has no connection, inside. Thus, the extreme SuperHyperSet of the extreme SuperHyperVertices with one SuperHyperElement has been ignored in the exploring to lead on the optimal case implying the extreme R-SuperHyperMatching. The extreme R-SuperHyperMatching with the exclusion of the exclusion of all  extreme SuperHyperVertices in one extreme SuperHyperEdge and with other terms, the extreme R-SuperHyperMatching with the inclusion of all extreme SuperHyperVertices in one extreme SuperHyperEdge, is an extreme quasi-R-SuperHyperMatching. To sum them up, in a connected non-obvious extreme  SuperHyperGraph $ESHG:(V,E).$ There's only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only the maximum possibilities of the distinct interior extreme SuperHyperVertices inside of any given extreme quasi-R-SuperHyperMatching minus all  extreme SuperHypeNeighbor to some of them but not all of them. In other words, there's only an unique extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only two distinct extreme SuperHyperVertices in an  extreme quasi-R-SuperHyperMatching, minus all  extreme SuperHypeNeighbor to some of them but not all of them.
\\
The main definition of the extreme R-SuperHyperMatching has two titles. An extreme quasi-R-SuperHyperMatching and its corresponded quasi-maximum extreme R-SuperHyperCardinality are two titles in the terms of quasi-R-styles. For any extreme number, there's an extreme quasi-R-SuperHyperMatching with that quasi-maximum extreme SuperHyperCardinality in the terms of the embedded extreme SuperHyperGraph. If there's an embedded extreme SuperHyperGraph, then the extreme quasi-SuperHyperNotions lead us to take the collection of all the extreme quasi-R-SuperHyperMatchings for all extreme numbers less than its extreme corresponded maximum number. The essence of the extreme SuperHyperMatching ends up but this essence starts up in the terms of the extreme quasi-R-SuperHyperMatching, again and more in the operations of collecting all the extreme quasi-R-SuperHyperMatchings acted on the all possible used formations of the extreme SuperHyperGraph to achieve one extreme number. This extreme number is considered as the equivalence class for all corresponded quasi-R-SuperHyperMatchings. Let $z_{\text{Extreme Number}},S_{\text{Extreme SuperHyperSet}}$ and $G_{\text{Extreme SuperHyperMatching}}$ be an extreme number, an extreme SuperHyperSet and an extreme SuperHyperMatching. Then
\begin{eqnarray*}
&&[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}\}.
\end{eqnarray*}
As its consequences, the formal definition of the extreme SuperHyperMatching is re-formalized and redefined as follows.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}\}.
\end{eqnarray*}
To get more precise perceptions, the follow-up expressions propose another formal technical definition for the extreme SuperHyperMatching.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
In more concise and more convenient ways, the modified definition for the extreme SuperHyperMatching poses the upcoming expressions.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
To translate the statement to this mathematical literature, the  formulae will be revised.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
And then,
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
To get more visions in the closer look-up, there's an overall overlook.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Now, the extension of these types of approaches is up. Since the new term, ``extreme SuperHyperNeighborhood'', could be redefined as the collection of the extreme SuperHyperVertices such that any amount of its extreme SuperHyperVertices are incident to an extreme  SuperHyperEdge. It's, literarily,  another name for ``extreme  Quasi-SuperHyperMatching'' but, precisely, it's the generalization of  ``extreme  Quasi-SuperHyperMatching'' since ``extreme Quasi-SuperHyperMatching'' happens ``extreme SuperHyperMatching'' in an extreme SuperHyperGraph as initial framework and background but ``extreme SuperHyperNeighborhood'' may not happens ``extreme SuperHyperMatching'' in an extreme SuperHyperGraph as initial framework and preliminarily background since there are some ambiguities about the extreme SuperHyperCardinality arise from it. To get orderly keywords, the terms, ``extreme SuperHyperNeighborhood'',  ``extreme Quasi-SuperHyperMatching'', and  ``extreme SuperHyperMatching'' are up.
\\
Thus, let $z_{\text{Extreme Number}},N_{\text{Extreme SuperHyperNeighborhood}}$ and $G_{\text{Extreme SuperHyperMatching}}$ be an extreme number, an extreme SuperHyperNeighborhood and an extreme SuperHyperMatching and the new terms are up.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
And with go back to initial structure,
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Thus, in a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\\
  To make sense with the precise words in the terms of ``R-', the follow-up illustrations are coming up.
  \\
  The following extreme SuperHyperSet  of extreme  SuperHyperVertices is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching.
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   The extreme SuperHyperSet of extreme SuperHyperVertices,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperVertices,  
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an \underline{\textbf{extreme R-SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge amid some extreme SuperHyperVertices instead of all given by \underline{\textbf{extreme SuperHyperMatching}} is related to the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   There's   \underline{not} only \underline{\textbf{one}} extreme SuperHyperVertex \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{one}} extreme SuperHyperVertex. But the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   doesn't have less than two  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet since they've come from at least so far an SuperHyperEdge. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperVertices,
 $$(V\setminus V\setminus \{x,z\})\cup\{xy\}$$ or $$(V\setminus V\setminus \{x,z\})\cup\{zy\}$$
   is an  extreme R-SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices instead of all given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme SuperHyperEdge for some amount extreme  SuperHyperVertices instead of all given by that extreme type-SuperHyperSet called the  extreme SuperHyperMatching. There isn't  only less than two extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   Thus the non-obvious  extreme R-SuperHyperMatching,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  is up. The non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is  the extreme SuperHyperSet, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 does includes only less than two SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E)$ but it's impossible in the case, they've corresponded to an SuperHyperEdge. It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  R-SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme R-SuperHyperMatching}},
\end{center}
 is only and only
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$  with a illustrated SuperHyperModeling. It's also, not only an extreme free-triangle embedded SuperHyperModel and an extreme on-triangle embedded SuperHyperModel but also it's an extreme stable embedded SuperHyperModel. But all only non-obvious simple extreme type-SuperHyperSets of the extreme  R-SuperHyperMatching amid those obvious simple extreme type-SuperHyperSets of the extreme SuperHyperMatching, are
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
\\
To sum them up,  assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 is an extreme R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of an extreme  R-SuperHyperMatching is the cardinality of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  \\
To sum them up,  in a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\\
Assume a connected extreme  SuperHyperGraph $ESHG:(V,E).$ Let an extreme SuperHyperEdge $ESHE: E\in E_{ESHG:(V,E)}$ has some extreme SuperHyperVertices $r.$ Consider all extreme numbers of those extreme SuperHyperVertices from that extreme SuperHyperEdge excluding excluding more than $r$ distinct extreme SuperHyperVertices, exclude to any given extreme SuperHyperSet of the extreme SuperHyperVertices. Consider there's an extreme  R-SuperHyperMatching with the least cardinality, the lower sharp extreme bound for extreme cardinality. Assume a connected extreme SuperHyperGraph $ESHG:(V,E).$ The extreme SuperHyperSet of the extreme SuperHyperVertices $ V_{ESHE}\setminus \{z\}$ is an extreme SuperHyperSet $S$ of  the extreme SuperHyperVertices such that there's an extreme SuperHyperEdge to have  some extreme SuperHyperVertices uniquely but it isn't an extreme R-SuperHyperMatching. Since it doesn't have  \underline{\textbf{the maximum extreme cardinality}} of an extreme SuperHyperSet $S$ of  extreme SuperHyperVertices such that there's an extreme SuperHyperEdge to have some SuperHyperVertices uniquely. The extreme SuperHyperSet of the extreme SuperHyperVertices $V_{ESHE}\cup \{z\}$ is the maximum extreme cardinality of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices but it isn't an extreme R-SuperHyperMatching. Since it \textbf{\underline{doesn't do}} the extreme procedure such that such that there's an extreme SuperHyperEdge to have some extreme  SuperHyperVertices uniquely  [there are at least one extreme SuperHyperVertex outside  implying there's, sometimes in  the connected extreme SuperHyperGraph $ESHG:(V,E),$ an extreme SuperHyperVertex, titled its extreme SuperHyperNeighbor,  to that extreme SuperHyperVertex in the extreme SuperHyperSet $S$ so as $S$ doesn't do ``the extreme procedure''.]. There's  only \textbf{\underline{one}} extreme SuperHyperVertex    \textbf{\underline{outside}} the intended extreme SuperHyperSet, $V_{ESHE}\cup \{z\},$ in the terms of extreme SuperHyperNeighborhood. Thus the obvious extreme R-SuperHyperMatching,  $V_{ESHE}$ is up. The obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching,  $V_{ESHE},$ \textbf{\underline{is}} an extreme SuperHyperSet, $V_{ESHE},$  \textbf{\underline{includes}} only \textbf{\underline{all}}  extreme SuperHyperVertices does forms any kind of extreme pairs are titled   \underline{extreme SuperHyperNeighbors} in a connected extreme SuperHyperGraph $ESHG:(V,E).$ Since the extreme SuperHyperSet of the extreme SuperHyperVertices $V_{ESHE},$ is the \textbf{\underline{maximum extreme SuperHyperCardinality}} of an extreme SuperHyperSet $S$ of  extreme SuperHyperVertices  \textbf{\underline{such that}}  there's an extreme SuperHyperEdge to have some extreme SuperHyperVertices uniquely. Thus, in a connected extreme SuperHyperGraph $ESHG:(V,E).$ Any extreme R-SuperHyperMatching only contains all interior extreme SuperHyperVertices and all exterior extreme SuperHyperVertices from the unique extreme SuperHyperEdge where there's any of them has all possible  extreme SuperHyperNeighbors in and there's all  extreme SuperHyperNeighborhoods in with no exception minus all  extreme SuperHypeNeighbors to some of them not all of them  but everything is possible about extreme SuperHyperNeighborhoods and extreme SuperHyperNeighbors out.
\\
The SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
 \end{proof}

 \begin{example}\label{118EXM21a}
In the extreme Figure \eqref{118NSHG21a}, the connected extreme SuperHyperBipartite $ESHB:(V,E),$ is extreme highlighted and extreme featured. The obtained extreme SuperHyperSet, by the extreme Algorithm in previous extreme result, of the extreme SuperHyperVertices of the connected extreme SuperHyperBipartite $ESHB:(V,E),$ in the extreme SuperHyperModel \eqref{118NSHG21a}, is the extreme  SuperHyperMatching.
\begin{figure}
\includegraphics[width=100mm]{118NSHG21.png}
 \caption{An extreme SuperHyperBipartite extreme Associated to the extreme Notions of  extreme SuperHyperMatching in the Example \eqref{118EXM21a}}
 \label{118NSHG21a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected extreme SuperHyperMultipartite $ESHM:(V,E).$ Then an extreme R-SuperHyperMatching is an extreme SuperHyperSet of the interior extreme SuperHyperVertices with only no extreme exception in the extreme form of interior extreme SuperHyperVertices from an extreme SuperHyperPart and only no  exception in the form of interior SuperHyperVertices from another SuperHyperPart  titled ``SuperHyperNeighbors'' with neglecting and ignoring more than some of them aren't SuperHyperNeighbors to all. An extreme  R-SuperHyperMatching has the extreme maximum number on  all the extreme summation on the extreme cardinality of the all extreme SuperHyperParts form some SuperHyperEdges minus those make extreme SuperHypeNeighbors to some not all or not unique. Also,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
Assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ The SuperHyperSet of the SuperHyperVertices $V\setminus V\setminus \{z\}$ isn't a quasi-R-SuperHyperMatching since neither amount of extreme SuperHyperEdges nor amount of SuperHyperVertices where amount refers to the extreme number of SuperHyperVertices(-/SuperHyperEdges) more than one to form any kind of SuperHyperEdges or any number of SuperHyperEdges. Let us consider the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
This extreme SuperHyperSet of the extreme SuperHyperVertices has the eligibilities to propose property such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices but the maximum extreme cardinality indicates that these extreme  type-SuperHyperSets couldn't give us the extreme lower bound in the term of extreme sharpness. In other words, the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
of the extreme SuperHyperVertices implies at least on-quasi-triangle style is up but sometimes the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
 of the extreme SuperHyperVertices is free-quasi-triangle and it doesn't make a contradiction to the supposition on the connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Thus the minimum case never happens in the generality of the connected loopless neutrosophic SuperHyperGraphs. Thus if we assume in the worst case, literally,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is a quasi-R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of a  quasi-R-SuperHyperMatching is the cardinality of
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 Then we've lost some connected loopless neutrosophic SuperHyperClasses of the connected loopless neutrosophic SuperHyperGraphs titled free-triangle, on-triangle, and their quasi-types but the SuperHyperStable is only up in this quasi-R-SuperHyperMatching. It's the contradiction to that fact on the generality. There are some counterexamples to deny this statement. One of them comes from the setting of the graph titled path and cycle as the counterexamples-classes or reversely direction star as the examples-classes, are well-known classes in that setting and they could be considered as the examples-classes and counterexamples-classes for the tight bound of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  Let $V\setminus V\setminus \{z\}$ in mind. There's no necessity on the SuperHyperEdge since we need at least two SuperHyperVertices to form a SuperHyperEdge. It doesn't withdraw the principles of the main definition since there's no condition to be satisfied but the condition is on the existence of the SuperHyperEdge instead of acting on the SuperHyperVertices. In other words, if there's a SuperHyperEdge, then the extreme SuperHyperSet has the necessary condition for the intended definition to be applied. Thus the $V\setminus V\setminus \{z\}$ is withdrawn not by the conditions of the main definition but by the necessity of the pre-condition on the usage of the main definition.
  \\
  The extreme structure of the extreme R-SuperHyperMatching decorates the extreme SuperHyperVertices don't have received any extreme connections so as this extreme style implies different versions of extreme SuperHyperEdges with the maximum extreme cardinality in the terms of extreme SuperHyperVertices are spotlight. The lower extreme bound is to have the maximum extreme groups of extreme SuperHyperVertices have perfect extreme connections inside each of SuperHyperEdges and the outside of this extreme SuperHyperSet doesn't matter but regarding the connectedness of the used extreme SuperHyperGraph arising from its extreme properties taken from the fact that it's simple. If there's no more than one extreme SuperHyperVertex in the targeted extreme SuperHyperSet, then there's no extreme connection. Furthermore, the extreme existence of one extreme SuperHyperVertex has no  extreme effect to talk about the extreme R-SuperHyperMatching. Since at least two extreme SuperHyperVertices involve to make a title in the extreme background of the extreme SuperHyperGraph. The extreme SuperHyperGraph is obvious if it has no extreme SuperHyperEdge but at least two extreme SuperHyperVertices make the extreme version of extreme SuperHyperEdge. Thus in the extreme setting of non-obvious extreme SuperHyperGraph, there are at least one extreme SuperHyperEdge. It's necessary to mention that the word ``Simple'' is used as extreme adjective for the initial extreme SuperHyperGraph, induces there's no extreme  appearance of the loop extreme version of the extreme SuperHyperEdge and this extreme SuperHyperGraph is said to be loopless. The extreme adjective ``loop'' on the basic extreme framework engages one extreme SuperHyperVertex but it never happens in this extreme setting. With these extreme bases, on an extreme SuperHyperGraph, there's at least one extreme SuperHyperEdge thus there's at least an extreme R-SuperHyperMatching has the extreme cardinality of an extreme SuperHyperEdge. Thus, an extreme R-SuperHyperMatching has the extreme cardinality at least an extreme SuperHyperEdge. Assume an extreme SuperHyperSet $V\setminus V\setminus \{z\}.$ This extreme SuperHyperSet isn't an extreme R-SuperHyperMatching since either the extreme SuperHyperGraph is an obvious extreme SuperHyperModel thus it never happens since there's no extreme usage of this extreme framework and even more there's no extreme connection inside or the extreme SuperHyperGraph isn't obvious and as its consequences, there's an extreme contradiction with the term ``extreme R-SuperHyperMatching'' since the maximum extreme cardinality never happens for this extreme style of the extreme SuperHyperSet and beyond that there's no extreme connection inside as mentioned in first extreme case in the forms of drawback for this selected extreme SuperHyperSet. Let
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Comes up. This extreme case implies having the extreme style of on-quasi-triangle extreme style on the every extreme elements of this extreme SuperHyperSet. Precisely, the extreme R-SuperHyperMatching is the extreme SuperHyperSet of the extreme SuperHyperVertices such that some extreme amount of the extreme SuperHyperVertices are on-quasi-triangle extreme style. The extreme cardinality of the v SuperHypeSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Is the maximum in comparison to the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But the lower extreme bound is up. Thus the minimum extreme cardinality of the maximum extreme cardinality ends up the extreme discussion. The first extreme term refers to the extreme setting of the extreme SuperHyperGraph but this key point is enough since there's an extreme SuperHyperClass of an extreme SuperHyperGraph has no on-quasi-triangle extreme style amid some amount of its extreme SuperHyperVertices. This extreme setting of the extreme SuperHyperModel proposes an extreme SuperHyperSet has only some amount  extreme SuperHyperVertices from one extreme SuperHyperEdge such that there's no extreme amount of extreme SuperHyperEdges more than one involving these some amount of these extreme SuperHyperVertices. The extreme cardinality of this extreme SuperHyperSet is the maximum and the extreme case is occurred in the minimum extreme situation. To sum them up, the extreme SuperHyperSet
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Has the maximum extreme cardinality such that
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Contains some extreme SuperHyperVertices such that there's distinct-covers-order-amount extreme SuperHyperEdges for amount of extreme SuperHyperVertices taken from the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 It means that the extreme SuperHyperSet of the extreme SuperHyperVertices
  $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an extreme  R-SuperHyperMatching for the extreme SuperHyperGraph as used extreme background in the extreme terms of worst extreme case and the common theme of the lower extreme bound occurred in the specific extreme SuperHyperClasses of the extreme SuperHyperGraphs which are extreme free-quasi-triangle.
  \\
Assume an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$  extreme number of the extreme SuperHyperVertices. Then every extreme SuperHyperVertex has at least no extreme SuperHyperEdge with others in common. Thus those extreme SuperHyperVertices have the eligibles to be contained in an extreme R-SuperHyperMatching. Those extreme SuperHyperVertices are potentially included in an extreme  style-R-SuperHyperMatching. Formally, consider
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
Are the extreme SuperHyperVertices of an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z.$$ where the $\sim$ isn't an equivalence relation but only the symmetric relation on the extreme  SuperHyperVertices of the extreme SuperHyperGraph. The formal definition is as follows. $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z$$ if and only if $Z_i$ and $Z_j$ are the extreme SuperHyperVertices and there's only and only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ between the extreme SuperHyperVertices $Z_i$ and $Z_j.$ The other definition for the extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ in the terms of extreme R-SuperHyperMatching is
$$\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}.$$
This definition coincides with the definition of the extreme R-SuperHyperMatching but with slightly differences in the maximum extreme cardinality amid those extreme type-SuperHyperSets of the extreme SuperHyperVertices. Thus the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$\max_z|\{Z_1,Z_2,\ldots,Z_z~|~Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}|_{\text{extreme cardinality}},$$
and
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
is formalized with mathematical literatures on the extreme R-SuperHyperMatching. Let $Z_i\overset E\sim Z_j,$ be defined as $Z_i$ and $Z_j$ are the extreme SuperHyperVertices belong to the extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus,  
$$E=\{Z_1,Z_2,\ldots,Z_z~|~Z_i\overset E\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}.$$
Or
$$\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But with the slightly differences,
\begin{eqnarray*}
 &&
 \text{extreme R-SuperHyperMatching}=
 \\&&
 \{Z_1,Z_2,\ldots,Z_z~|~\forall i\neq j,~ i,j=1,2,\ldots,z,~\exists E_x,~Z_i\overset{E_x}\sim Z_j,\}.
\end{eqnarray*}
\begin{eqnarray*}
 &&
 \text{extreme R-SuperHyperMatching}=
 \\&&
V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.
\end{eqnarray*}
Thus $E\in E_{ESHG:(V,E)}$ is an extreme quasi-R-SuperHyperMatching where $E\in E_{ESHG:(V,E)}$ is fixed that means $E_x=E\in E_{ESHG:(V,E)}.$ for all extreme intended SuperHyperVertices but in an extreme SuperHyperMatching, $E_x=E\in E_{ESHG:(V,E)}$ could be different and it's not unique. To sum them up, in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ If an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$ extreme SuperHyperVertices, then the extreme cardinality of the extreme R-SuperHyperMatching is at least
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
 It's straightforward that  the extreme cardinality of the extreme R-SuperHyperMatching is at least the maximum extreme number of extreme SuperHyperVertices of the extreme SuperHyperEdges with the maximum number of the extreme SuperHyperEdges. In other words, the maximum number of the extreme SuperHyperEdges contains the maximum extreme number of extreme SuperHyperVertices are renamed to extreme SuperHyperMatching in some cases but the maximum number of the  extreme SuperHyperEdge with the maximum extreme number of extreme SuperHyperVertices, has the extreme SuperHyperVertices are contained in an extreme R-SuperHyperMatching.
\\
The obvious SuperHyperGraph has no extreme SuperHyperEdges. But the non-obvious extreme SuperHyperModel is up. The quasi-SuperHyperModel addresses some issues about the extreme optimal SuperHyperObject. It specially delivers some remarks on the extreme SuperHyperSet of the extreme SuperHyperVertices such that there's distinct amount of extreme SuperHyperEdges for distinct amount of extreme SuperHyperVertices up to all  taken from that extreme SuperHyperSet of the extreme SuperHyperVertices but this extreme SuperHyperSet of the extreme SuperHyperVertices is either has the maximum extreme SuperHyperCardinality or it doesn't have  maximum extreme SuperHyperCardinality. In a non-obvious SuperHyperModel, there's at least one extreme SuperHyperEdge containing at least all extreme SuperHyperVertices. Thus it forms an extreme quasi-R-SuperHyperMatching where the extreme completion of the extreme incidence is up in that.  Thus it's, literarily, an extreme embedded R-SuperHyperMatching. The SuperHyperNotions of embedded SuperHyperSet and quasi-SuperHyperSet coincide. In the original setting, these types of SuperHyperSets only don't satisfy on the maximum SuperHyperCardinality. Thus the embedded setting is elected such that those SuperHyperSets have the maximum extreme SuperHyperCardinality and they're extreme SuperHyperOptimal. The less than two distinct types of extreme SuperHyperVertices are included in the minimum extreme style of the embedded extreme R-SuperHyperMatching. The interior types of the extreme SuperHyperVertices are deciders. Since the extreme number of SuperHyperNeighbors are only  affected by the interior extreme SuperHyperVertices. The common connections, more precise and more formal, the perfect unique connections inside the extreme SuperHyperSet for any distinct types of extreme SuperHyperVertices pose the extreme R-SuperHyperMatching. Thus extreme exterior SuperHyperVertices could be used only in one extreme SuperHyperEdge and in extreme SuperHyperRelation with the interior extreme SuperHyperVertices in that  extreme SuperHyperEdge. In the embedded extreme SuperHyperMatching, there's the usage of exterior extreme SuperHyperVertices since they've more connections inside more than outside. Thus the title ``exterior'' is more relevant than the title ``interior''. One extreme SuperHyperVertex has no connection, inside. Thus, the extreme SuperHyperSet of the extreme SuperHyperVertices with one SuperHyperElement has been ignored in the exploring to lead on the optimal case implying the extreme R-SuperHyperMatching. The extreme R-SuperHyperMatching with the exclusion of the exclusion of all  extreme SuperHyperVertices in one extreme SuperHyperEdge and with other terms, the extreme R-SuperHyperMatching with the inclusion of all extreme SuperHyperVertices in one extreme SuperHyperEdge, is an extreme quasi-R-SuperHyperMatching. To sum them up, in a connected non-obvious extreme  SuperHyperGraph $ESHG:(V,E).$ There's only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only the maximum possibilities of the distinct interior extreme SuperHyperVertices inside of any given extreme quasi-R-SuperHyperMatching minus all  extreme SuperHypeNeighbor to some of them but not all of them. In other words, there's only an unique extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only two distinct extreme SuperHyperVertices in an  extreme quasi-R-SuperHyperMatching, minus all  extreme SuperHypeNeighbor to some of them but not all of them.
\\
The main definition of the extreme R-SuperHyperMatching has two titles. An extreme quasi-R-SuperHyperMatching and its corresponded quasi-maximum extreme R-SuperHyperCardinality are two titles in the terms of quasi-R-styles. For any extreme number, there's an extreme quasi-R-SuperHyperMatching with that quasi-maximum extreme SuperHyperCardinality in the terms of the embedded extreme SuperHyperGraph. If there's an embedded extreme SuperHyperGraph, then the extreme quasi-SuperHyperNotions lead us to take the collection of all the extreme quasi-R-SuperHyperMatchings for all extreme numbers less than its extreme corresponded maximum number. The essence of the extreme SuperHyperMatching ends up but this essence starts up in the terms of the extreme quasi-R-SuperHyperMatching, again and more in the operations of collecting all the extreme quasi-R-SuperHyperMatchings acted on the all possible used formations of the extreme SuperHyperGraph to achieve one extreme number. This extreme number is considered as the equivalence class for all corresponded quasi-R-SuperHyperMatchings. Let $z_{\text{Extreme Number}},S_{\text{Extreme SuperHyperSet}}$ and $G_{\text{Extreme SuperHyperMatching}}$ be an extreme number, an extreme SuperHyperSet and an extreme SuperHyperMatching. Then
\begin{eqnarray*}
&&[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}\}.
\end{eqnarray*}
As its consequences, the formal definition of the extreme SuperHyperMatching is re-formalized and redefined as follows.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}\}.
\end{eqnarray*}
To get more precise perceptions, the follow-up expressions propose another formal technical definition for the extreme SuperHyperMatching.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
In more concise and more convenient ways, the modified definition for the extreme SuperHyperMatching poses the upcoming expressions.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
To translate the statement to this mathematical literature, the  formulae will be revised.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
And then,
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
To get more visions in the closer look-up, there's an overall overlook.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Now, the extension of these types of approaches is up. Since the new term, ``extreme SuperHyperNeighborhood'', could be redefined as the collection of the extreme SuperHyperVertices such that any amount of its extreme SuperHyperVertices are incident to an extreme  SuperHyperEdge. It's, literarily,  another name for ``extreme  Quasi-SuperHyperMatching'' but, precisely, it's the generalization of  ``extreme  Quasi-SuperHyperMatching'' since ``extreme Quasi-SuperHyperMatching'' happens ``extreme SuperHyperMatching'' in an extreme SuperHyperGraph as initial framework and background but ``extreme SuperHyperNeighborhood'' may not happens ``extreme SuperHyperMatching'' in an extreme SuperHyperGraph as initial framework and preliminarily background since there are some ambiguities about the extreme SuperHyperCardinality arise from it. To get orderly keywords, the terms, ``extreme SuperHyperNeighborhood'',  ``extreme Quasi-SuperHyperMatching'', and  ``extreme SuperHyperMatching'' are up.
\\
Thus, let $z_{\text{Extreme Number}},N_{\text{Extreme SuperHyperNeighborhood}}$ and $G_{\text{Extreme SuperHyperMatching}}$ be an extreme number, an extreme SuperHyperNeighborhood and an extreme SuperHyperMatching and the new terms are up.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
And with go back to initial structure,
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Thus, in a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\\
  To make sense with the precise words in the terms of ``R-', the follow-up illustrations are coming up.
  \\
  The following extreme SuperHyperSet  of extreme  SuperHyperVertices is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching.
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   The extreme SuperHyperSet of extreme SuperHyperVertices,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperVertices,  
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an \underline{\textbf{extreme R-SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge amid some extreme SuperHyperVertices instead of all given by \underline{\textbf{extreme SuperHyperMatching}} is related to the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   There's   \underline{not} only \underline{\textbf{one}} extreme SuperHyperVertex \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{one}} extreme SuperHyperVertex. But the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   doesn't have less than two  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet since they've come from at least so far an SuperHyperEdge. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperVertices,
 $$(V\setminus V\setminus \{x,z\})\cup\{xy\}$$ or $$(V\setminus V\setminus \{x,z\})\cup\{zy\}$$
   is an  extreme R-SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices instead of all given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme SuperHyperEdge for some amount extreme  SuperHyperVertices instead of all given by that extreme type-SuperHyperSet called the  extreme SuperHyperMatching. There isn't  only less than two extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   Thus the non-obvious  extreme R-SuperHyperMatching,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  is up. The non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is  the extreme SuperHyperSet, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 does includes only less than two SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E)$ but it's impossible in the case, they've corresponded to an SuperHyperEdge. It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  R-SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme R-SuperHyperMatching}},
\end{center}
 is only and only
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$  with a illustrated SuperHyperModeling. It's also, not only an extreme free-triangle embedded SuperHyperModel and an extreme on-triangle embedded SuperHyperModel but also it's an extreme stable embedded SuperHyperModel. But all only non-obvious simple extreme type-SuperHyperSets of the extreme  R-SuperHyperMatching amid those obvious simple extreme type-SuperHyperSets of the extreme SuperHyperMatching, are
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
\\
To sum them up,  assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 is an extreme R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of an extreme  R-SuperHyperMatching is the cardinality of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  \\
To sum them up,  in a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\\
Assume a connected extreme  SuperHyperGraph $ESHG:(V,E).$ Let an extreme SuperHyperEdge $ESHE: E\in E_{ESHG:(V,E)}$ has some extreme SuperHyperVertices $r.$ Consider all extreme numbers of those extreme SuperHyperVertices from that extreme SuperHyperEdge excluding excluding more than $r$ distinct extreme SuperHyperVertices, exclude to any given extreme SuperHyperSet of the extreme SuperHyperVertices. Consider there's an extreme  R-SuperHyperMatching with the least cardinality, the lower sharp extreme bound for extreme cardinality. Assume a connected extreme SuperHyperGraph $ESHG:(V,E).$ The extreme SuperHyperSet of the extreme SuperHyperVertices $ V_{ESHE}\setminus \{z\}$ is an extreme SuperHyperSet $S$ of  the extreme SuperHyperVertices such that there's an extreme SuperHyperEdge to have  some extreme SuperHyperVertices uniquely but it isn't an extreme R-SuperHyperMatching. Since it doesn't have  \underline{\textbf{the maximum extreme cardinality}} of an extreme SuperHyperSet $S$ of  extreme SuperHyperVertices such that there's an extreme SuperHyperEdge to have some SuperHyperVertices uniquely. The extreme SuperHyperSet of the extreme SuperHyperVertices $V_{ESHE}\cup \{z\}$ is the maximum extreme cardinality of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices but it isn't an extreme R-SuperHyperMatching. Since it \textbf{\underline{doesn't do}} the extreme procedure such that such that there's an extreme SuperHyperEdge to have some extreme  SuperHyperVertices uniquely  [there are at least one extreme SuperHyperVertex outside  implying there's, sometimes in  the connected extreme SuperHyperGraph $ESHG:(V,E),$ an extreme SuperHyperVertex, titled its extreme SuperHyperNeighbor,  to that extreme SuperHyperVertex in the extreme SuperHyperSet $S$ so as $S$ doesn't do ``the extreme procedure''.]. There's  only \textbf{\underline{one}} extreme SuperHyperVertex    \textbf{\underline{outside}} the intended extreme SuperHyperSet, $V_{ESHE}\cup \{z\},$ in the terms of extreme SuperHyperNeighborhood. Thus the obvious extreme R-SuperHyperMatching,  $V_{ESHE}$ is up. The obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching,  $V_{ESHE},$ \textbf{\underline{is}} an extreme SuperHyperSet, $V_{ESHE},$  \textbf{\underline{includes}} only \textbf{\underline{all}}  extreme SuperHyperVertices does forms any kind of extreme pairs are titled   \underline{extreme SuperHyperNeighbors} in a connected extreme SuperHyperGraph $ESHG:(V,E).$ Since the extreme SuperHyperSet of the extreme SuperHyperVertices $V_{ESHE},$ is the \textbf{\underline{maximum extreme SuperHyperCardinality}} of an extreme SuperHyperSet $S$ of  extreme SuperHyperVertices  \textbf{\underline{such that}}  there's an extreme SuperHyperEdge to have some extreme SuperHyperVertices uniquely. Thus, in a connected extreme SuperHyperGraph $ESHG:(V,E).$ Any extreme R-SuperHyperMatching only contains all interior extreme SuperHyperVertices and all exterior extreme SuperHyperVertices from the unique extreme SuperHyperEdge where there's any of them has all possible  extreme SuperHyperNeighbors in and there's all  extreme SuperHyperNeighborhoods in with no exception minus all  extreme SuperHypeNeighbors to some of them not all of them  but everything is possible about extreme SuperHyperNeighborhoods and extreme SuperHyperNeighbors out.
\\
The SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=z^{\min|P_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}. \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}.
\end{eqnarray*}
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
 \end{proof}

 \begin{example}\label{118EXM22a}
In the Figure \eqref{118NSHG22a}, the connected extreme SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and extreme featured. The obtained extreme SuperHyperSet, by the Algorithm in previous extreme result, of the extreme SuperHyperVertices of the connected extreme SuperHyperMultipartite $ESHM:(V,E),$ in the extreme SuperHyperModel \eqref{118NSHG22a}, is the  extreme SuperHyperMatching.
\begin{figure}
\includegraphics[width=100mm]{118NSHG22.png}
 \caption{An extreme  SuperHyperMultipartite Associated to the Notions of  extreme SuperHyperMatching in the Example \eqref{118EXM22a}}
 \label{118NSHG22a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected   extreme SuperHyperWheel $ESHW:(V,E).$ Then an extreme  R-SuperHyperMatching is an extreme SuperHyperSet of the interior extreme SuperHyperVertices,  excluding the extreme SuperHyperCenter, with only no  exception in the form of interior extreme SuperHyperVertices from same extreme SuperHyperEdge with the exclusion on extreme SuperHypeNeighbors to some of them and not all. An extreme  R-SuperHyperMatching has the extreme maximum number on all the extreme number of all the extreme SuperHyperEdges don't have common extreme SuperHyperNeighbors.  Also,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
Assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ The SuperHyperSet of the SuperHyperVertices $V\setminus V\setminus \{z\}$ isn't a quasi-R-SuperHyperMatching since neither amount of extreme SuperHyperEdges nor amount of SuperHyperVertices where amount refers to the extreme number of SuperHyperVertices(-/SuperHyperEdges) more than one to form any kind of SuperHyperEdges or any number of SuperHyperEdges. Let us consider the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
This extreme SuperHyperSet of the extreme SuperHyperVertices has the eligibilities to propose property such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices but the maximum extreme cardinality indicates that these extreme  type-SuperHyperSets couldn't give us the extreme lower bound in the term of extreme sharpness. In other words, the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
of the extreme SuperHyperVertices implies at least on-quasi-triangle style is up but sometimes the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
 of the extreme SuperHyperVertices is free-quasi-triangle and it doesn't make a contradiction to the supposition on the connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Thus the minimum case never happens in the generality of the connected loopless neutrosophic SuperHyperGraphs. Thus if we assume in the worst case, literally,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is a quasi-R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of a  quasi-R-SuperHyperMatching is the cardinality of
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 Then we've lost some connected loopless neutrosophic SuperHyperClasses of the connected loopless neutrosophic SuperHyperGraphs titled free-triangle, on-triangle, and their quasi-types but the SuperHyperStable is only up in this quasi-R-SuperHyperMatching. It's the contradiction to that fact on the generality. There are some counterexamples to deny this statement. One of them comes from the setting of the graph titled path and cycle as the counterexamples-classes or reversely direction star as the examples-classes, are well-known classes in that setting and they could be considered as the examples-classes and counterexamples-classes for the tight bound of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  Let $V\setminus V\setminus \{z\}$ in mind. There's no necessity on the SuperHyperEdge since we need at least two SuperHyperVertices to form a SuperHyperEdge. It doesn't withdraw the principles of the main definition since there's no condition to be satisfied but the condition is on the existence of the SuperHyperEdge instead of acting on the SuperHyperVertices. In other words, if there's a SuperHyperEdge, then the extreme SuperHyperSet has the necessary condition for the intended definition to be applied. Thus the $V\setminus V\setminus \{z\}$ is withdrawn not by the conditions of the main definition but by the necessity of the pre-condition on the usage of the main definition.
  \\
  The extreme structure of the extreme R-SuperHyperMatching decorates the extreme SuperHyperVertices don't have received any extreme connections so as this extreme style implies different versions of extreme SuperHyperEdges with the maximum extreme cardinality in the terms of extreme SuperHyperVertices are spotlight. The lower extreme bound is to have the maximum extreme groups of extreme SuperHyperVertices have perfect extreme connections inside each of SuperHyperEdges and the outside of this extreme SuperHyperSet doesn't matter but regarding the connectedness of the used extreme SuperHyperGraph arising from its extreme properties taken from the fact that it's simple. If there's no more than one extreme SuperHyperVertex in the targeted extreme SuperHyperSet, then there's no extreme connection. Furthermore, the extreme existence of one extreme SuperHyperVertex has no  extreme effect to talk about the extreme R-SuperHyperMatching. Since at least two extreme SuperHyperVertices involve to make a title in the extreme background of the extreme SuperHyperGraph. The extreme SuperHyperGraph is obvious if it has no extreme SuperHyperEdge but at least two extreme SuperHyperVertices make the extreme version of extreme SuperHyperEdge. Thus in the extreme setting of non-obvious extreme SuperHyperGraph, there are at least one extreme SuperHyperEdge. It's necessary to mention that the word ``Simple'' is used as extreme adjective for the initial extreme SuperHyperGraph, induces there's no extreme  appearance of the loop extreme version of the extreme SuperHyperEdge and this extreme SuperHyperGraph is said to be loopless. The extreme adjective ``loop'' on the basic extreme framework engages one extreme SuperHyperVertex but it never happens in this extreme setting. With these extreme bases, on an extreme SuperHyperGraph, there's at least one extreme SuperHyperEdge thus there's at least an extreme R-SuperHyperMatching has the extreme cardinality of an extreme SuperHyperEdge. Thus, an extreme R-SuperHyperMatching has the extreme cardinality at least an extreme SuperHyperEdge. Assume an extreme SuperHyperSet $V\setminus V\setminus \{z\}.$ This extreme SuperHyperSet isn't an extreme R-SuperHyperMatching since either the extreme SuperHyperGraph is an obvious extreme SuperHyperModel thus it never happens since there's no extreme usage of this extreme framework and even more there's no extreme connection inside or the extreme SuperHyperGraph isn't obvious and as its consequences, there's an extreme contradiction with the term ``extreme R-SuperHyperMatching'' since the maximum extreme cardinality never happens for this extreme style of the extreme SuperHyperSet and beyond that there's no extreme connection inside as mentioned in first extreme case in the forms of drawback for this selected extreme SuperHyperSet. Let
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Comes up. This extreme case implies having the extreme style of on-quasi-triangle extreme style on the every extreme elements of this extreme SuperHyperSet. Precisely, the extreme R-SuperHyperMatching is the extreme SuperHyperSet of the extreme SuperHyperVertices such that some extreme amount of the extreme SuperHyperVertices are on-quasi-triangle extreme style. The extreme cardinality of the v SuperHypeSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Is the maximum in comparison to the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But the lower extreme bound is up. Thus the minimum extreme cardinality of the maximum extreme cardinality ends up the extreme discussion. The first extreme term refers to the extreme setting of the extreme SuperHyperGraph but this key point is enough since there's an extreme SuperHyperClass of an extreme SuperHyperGraph has no on-quasi-triangle extreme style amid some amount of its extreme SuperHyperVertices. This extreme setting of the extreme SuperHyperModel proposes an extreme SuperHyperSet has only some amount  extreme SuperHyperVertices from one extreme SuperHyperEdge such that there's no extreme amount of extreme SuperHyperEdges more than one involving these some amount of these extreme SuperHyperVertices. The extreme cardinality of this extreme SuperHyperSet is the maximum and the extreme case is occurred in the minimum extreme situation. To sum them up, the extreme SuperHyperSet
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Has the maximum extreme cardinality such that
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Contains some extreme SuperHyperVertices such that there's distinct-covers-order-amount extreme SuperHyperEdges for amount of extreme SuperHyperVertices taken from the extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 It means that the extreme SuperHyperSet of the extreme SuperHyperVertices
  $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an extreme  R-SuperHyperMatching for the extreme SuperHyperGraph as used extreme background in the extreme terms of worst extreme case and the common theme of the lower extreme bound occurred in the specific extreme SuperHyperClasses of the extreme SuperHyperGraphs which are extreme free-quasi-triangle.
  \\
Assume an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$  extreme number of the extreme SuperHyperVertices. Then every extreme SuperHyperVertex has at least no extreme SuperHyperEdge with others in common. Thus those extreme SuperHyperVertices have the eligibles to be contained in an extreme R-SuperHyperMatching. Those extreme SuperHyperVertices are potentially included in an extreme  style-R-SuperHyperMatching. Formally, consider
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
Are the extreme SuperHyperVertices of an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z.$$ where the $\sim$ isn't an equivalence relation but only the symmetric relation on the extreme  SuperHyperVertices of the extreme SuperHyperGraph. The formal definition is as follows. $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z$$ if and only if $Z_i$ and $Z_j$ are the extreme SuperHyperVertices and there's only and only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ between the extreme SuperHyperVertices $Z_i$ and $Z_j.$ The other definition for the extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ in the terms of extreme R-SuperHyperMatching is
$$\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}.$$
This definition coincides with the definition of the extreme R-SuperHyperMatching but with slightly differences in the maximum extreme cardinality amid those extreme type-SuperHyperSets of the extreme SuperHyperVertices. Thus the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$\max_z|\{Z_1,Z_2,\ldots,Z_z~|~Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}|_{\text{extreme cardinality}},$$
and
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
is formalized with mathematical literatures on the extreme R-SuperHyperMatching. Let $Z_i\overset E\sim Z_j,$ be defined as $Z_i$ and $Z_j$ are the extreme SuperHyperVertices belong to the extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus,  
$$E=\{Z_1,Z_2,\ldots,Z_z~|~Z_i\overset E\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}.$$
Or
$$\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But with the slightly differences,
\begin{eqnarray*}
 &&
 \text{extreme R-SuperHyperMatching}=
 \\&&
 \{Z_1,Z_2,\ldots,Z_z~|~\forall i\neq j,~ i,j=1,2,\ldots,z,~\exists E_x,~Z_i\overset{E_x}\sim Z_j,\}.
\end{eqnarray*}
\begin{eqnarray*}
 &&
 \text{extreme R-SuperHyperMatching}=
 \\&&
V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.
\end{eqnarray*}
Thus $E\in E_{ESHG:(V,E)}$ is an extreme quasi-R-SuperHyperMatching where $E\in E_{ESHG:(V,E)}$ is fixed that means $E_x=E\in E_{ESHG:(V,E)}.$ for all extreme intended SuperHyperVertices but in an extreme SuperHyperMatching, $E_x=E\in E_{ESHG:(V,E)}$ could be different and it's not unique. To sum them up, in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ If an extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$ extreme SuperHyperVertices, then the extreme cardinality of the extreme R-SuperHyperMatching is at least
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
 It's straightforward that  the extreme cardinality of the extreme R-SuperHyperMatching is at least the maximum extreme number of extreme SuperHyperVertices of the extreme SuperHyperEdges with the maximum number of the extreme SuperHyperEdges. In other words, the maximum number of the extreme SuperHyperEdges contains the maximum extreme number of extreme SuperHyperVertices are renamed to extreme SuperHyperMatching in some cases but the maximum number of the  extreme SuperHyperEdge with the maximum extreme number of extreme SuperHyperVertices, has the extreme SuperHyperVertices are contained in an extreme R-SuperHyperMatching.
\\
The obvious SuperHyperGraph has no extreme SuperHyperEdges. But the non-obvious extreme SuperHyperModel is up. The quasi-SuperHyperModel addresses some issues about the extreme optimal SuperHyperObject. It specially delivers some remarks on the extreme SuperHyperSet of the extreme SuperHyperVertices such that there's distinct amount of extreme SuperHyperEdges for distinct amount of extreme SuperHyperVertices up to all  taken from that extreme SuperHyperSet of the extreme SuperHyperVertices but this extreme SuperHyperSet of the extreme SuperHyperVertices is either has the maximum extreme SuperHyperCardinality or it doesn't have  maximum extreme SuperHyperCardinality. In a non-obvious SuperHyperModel, there's at least one extreme SuperHyperEdge containing at least all extreme SuperHyperVertices. Thus it forms an extreme quasi-R-SuperHyperMatching where the extreme completion of the extreme incidence is up in that.  Thus it's, literarily, an extreme embedded R-SuperHyperMatching. The SuperHyperNotions of embedded SuperHyperSet and quasi-SuperHyperSet coincide. In the original setting, these types of SuperHyperSets only don't satisfy on the maximum SuperHyperCardinality. Thus the embedded setting is elected such that those SuperHyperSets have the maximum extreme SuperHyperCardinality and they're extreme SuperHyperOptimal. The less than two distinct types of extreme SuperHyperVertices are included in the minimum extreme style of the embedded extreme R-SuperHyperMatching. The interior types of the extreme SuperHyperVertices are deciders. Since the extreme number of SuperHyperNeighbors are only  affected by the interior extreme SuperHyperVertices. The common connections, more precise and more formal, the perfect unique connections inside the extreme SuperHyperSet for any distinct types of extreme SuperHyperVertices pose the extreme R-SuperHyperMatching. Thus extreme exterior SuperHyperVertices could be used only in one extreme SuperHyperEdge and in extreme SuperHyperRelation with the interior extreme SuperHyperVertices in that  extreme SuperHyperEdge. In the embedded extreme SuperHyperMatching, there's the usage of exterior extreme SuperHyperVertices since they've more connections inside more than outside. Thus the title ``exterior'' is more relevant than the title ``interior''. One extreme SuperHyperVertex has no connection, inside. Thus, the extreme SuperHyperSet of the extreme SuperHyperVertices with one SuperHyperElement has been ignored in the exploring to lead on the optimal case implying the extreme R-SuperHyperMatching. The extreme R-SuperHyperMatching with the exclusion of the exclusion of all  extreme SuperHyperVertices in one extreme SuperHyperEdge and with other terms, the extreme R-SuperHyperMatching with the inclusion of all extreme SuperHyperVertices in one extreme SuperHyperEdge, is an extreme quasi-R-SuperHyperMatching. To sum them up, in a connected non-obvious extreme  SuperHyperGraph $ESHG:(V,E).$ There's only one  extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only the maximum possibilities of the distinct interior extreme SuperHyperVertices inside of any given extreme quasi-R-SuperHyperMatching minus all  extreme SuperHypeNeighbor to some of them but not all of them. In other words, there's only an unique extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only two distinct extreme SuperHyperVertices in an  extreme quasi-R-SuperHyperMatching, minus all  extreme SuperHypeNeighbor to some of them but not all of them.
\\
The main definition of the extreme R-SuperHyperMatching has two titles. An extreme quasi-R-SuperHyperMatching and its corresponded quasi-maximum extreme R-SuperHyperCardinality are two titles in the terms of quasi-R-styles. For any extreme number, there's an extreme quasi-R-SuperHyperMatching with that quasi-maximum extreme SuperHyperCardinality in the terms of the embedded extreme SuperHyperGraph. If there's an embedded extreme SuperHyperGraph, then the extreme quasi-SuperHyperNotions lead us to take the collection of all the extreme quasi-R-SuperHyperMatchings for all extreme numbers less than its extreme corresponded maximum number. The essence of the extreme SuperHyperMatching ends up but this essence starts up in the terms of the extreme quasi-R-SuperHyperMatching, again and more in the operations of collecting all the extreme quasi-R-SuperHyperMatchings acted on the all possible used formations of the extreme SuperHyperGraph to achieve one extreme number. This extreme number is considered as the equivalence class for all corresponded quasi-R-SuperHyperMatchings. Let $z_{\text{Extreme Number}},S_{\text{Extreme SuperHyperSet}}$ and $G_{\text{Extreme SuperHyperMatching}}$ be an extreme number, an extreme SuperHyperSet and an extreme SuperHyperMatching. Then
\begin{eqnarray*}
&&[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}\}.
\end{eqnarray*}
As its consequences, the formal definition of the extreme SuperHyperMatching is re-formalized and redefined as follows.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}\}.
\end{eqnarray*}
To get more precise perceptions, the follow-up expressions propose another formal technical definition for the extreme SuperHyperMatching.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
In more concise and more convenient ways, the modified definition for the extreme SuperHyperMatching poses the upcoming expressions.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
To translate the statement to this mathematical literature, the  formulae will be revised.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
And then,
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
To get more visions in the closer look-up, there's an overall overlook.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{S_{\text{Extreme SuperHyperSet}}~|
\\&&~S_{\text{Extreme SuperHyperSet}}=G_{\text{Extreme SuperHyperMatching}},
\\&&~|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{S\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|S_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Now, the extension of these types of approaches is up. Since the new term, ``extreme SuperHyperNeighborhood'', could be redefined as the collection of the extreme SuperHyperVertices such that any amount of its extreme SuperHyperVertices are incident to an extreme  SuperHyperEdge. It's, literarily,  another name for ``extreme  Quasi-SuperHyperMatching'' but, precisely, it's the generalization of  ``extreme  Quasi-SuperHyperMatching'' since ``extreme Quasi-SuperHyperMatching'' happens ``extreme SuperHyperMatching'' in an extreme SuperHyperGraph as initial framework and background but ``extreme SuperHyperNeighborhood'' may not happens ``extreme SuperHyperMatching'' in an extreme SuperHyperGraph as initial framework and preliminarily background since there are some ambiguities about the extreme SuperHyperCardinality arise from it. To get orderly keywords, the terms, ``extreme SuperHyperNeighborhood'',  ``extreme Quasi-SuperHyperMatching'', and  ``extreme SuperHyperMatching'' are up.
\\
Thus, let $z_{\text{Extreme Number}},N_{\text{Extreme SuperHyperNeighborhood}}$ and $G_{\text{Extreme SuperHyperMatching}}$ be an extreme number, an extreme SuperHyperNeighborhood and an extreme SuperHyperMatching and the new terms are up.
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\}.
\end{eqnarray*}
And with go back to initial structure,
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}=
\\&&
\cup_{z_{\text{Extreme Number}}}\{N_{\text{Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&=z_{\text{Extreme Number}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperNeighborhood}}|_{\text{Extreme Cardinality}}
\\&&
=\max_{[z_{\text{Extreme Number}}]_{\text{Extreme Class}}}z_{\text{Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{Extreme SuperHyperMatching}}=
\\&&
\{N_{\text{Extreme SuperHyperNeighborhood}}\in\cup_{z_{\text{Extreme Number}}}[z_{\text{Extreme Number}}]_{\text{Extreme Class}}~|~
\\&&
|N_{\text{Extreme SuperHyperSet}}|_{\text{Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Thus, in a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\\
  To make sense with the precise words in the terms of ``R-', the follow-up illustrations are coming up.
  \\
  The following extreme SuperHyperSet  of extreme  SuperHyperVertices is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching.
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   The extreme SuperHyperSet of extreme SuperHyperVertices,
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the simple extreme type-SuperHyperSet of the extreme R-SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperVertices,  
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an \underline{\textbf{extreme R-SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge amid some extreme SuperHyperVertices instead of all given by \underline{\textbf{extreme SuperHyperMatching}} is related to the extreme SuperHyperSet of the extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   There's   \underline{not} only \underline{\textbf{one}} extreme SuperHyperVertex \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{one}} extreme SuperHyperVertex. But the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   doesn't have less than two  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet since they've come from at least so far an SuperHyperEdge. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   \underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperVertices,
 $$(V\setminus V\setminus \{x,z\})\cup\{xy\}$$ or $$(V\setminus V\setminus \{x,z\})\cup\{zy\}$$
   is an  extreme R-SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices instead of all given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme SuperHyperEdge for some amount extreme  SuperHyperVertices instead of all given by that extreme type-SuperHyperSet called the  extreme SuperHyperMatching. There isn't  only less than two extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
   Thus the non-obvious  extreme R-SuperHyperMatching,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  is up. The non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is  the extreme SuperHyperSet, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 does includes only less than two SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E)$ but it's impossible in the case, they've corresponded to an SuperHyperEdge. It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  R-SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme R-SuperHyperMatching}},
\end{center}
 is only and only
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected neutrosophic SuperHyperGraph $ESHG:(V,E)$  with a illustrated SuperHyperModeling. It's also, not only an extreme free-triangle embedded SuperHyperModel and an extreme on-triangle embedded SuperHyperModel but also it's an extreme stable embedded SuperHyperModel. But all only non-obvious simple extreme type-SuperHyperSets of the extreme  R-SuperHyperMatching amid those obvious simple extreme type-SuperHyperSets of the extreme SuperHyperMatching, are
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
\\
To sum them up,  assume a connected loopless neutrosophic SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
 is an extreme R-SuperHyperMatching. In other words, the least cardinality, the lower sharp bound for the cardinality, of an extreme  R-SuperHyperMatching is the cardinality of
 $$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
  \\
To sum them up,  in a connected extreme SuperHyperGraph $ESHG:(V,E).$ The all interior extreme SuperHyperVertices belong to any extreme quasi-R-SuperHyperMatching if for any of them, and any of other corresponded extreme SuperHyperVertex, some interior extreme SuperHyperVertices are mutually extreme  SuperHyperNeighbors with no extreme exception at all minus  all extreme SuperHypeNeighbors to any amount of them.
\\
Assume a connected extreme  SuperHyperGraph $ESHG:(V,E).$ Let an extreme SuperHyperEdge $ESHE: E\in E_{ESHG:(V,E)}$ has some extreme SuperHyperVertices $r.$ Consider all extreme numbers of those extreme SuperHyperVertices from that extreme SuperHyperEdge excluding excluding more than $r$ distinct extreme SuperHyperVertices, exclude to any given extreme SuperHyperSet of the extreme SuperHyperVertices. Consider there's an extreme  R-SuperHyperMatching with the least cardinality, the lower sharp extreme bound for extreme cardinality. Assume a connected extreme SuperHyperGraph $ESHG:(V,E).$ The extreme SuperHyperSet of the extreme SuperHyperVertices $ V_{ESHE}\setminus \{z\}$ is an extreme SuperHyperSet $S$ of  the extreme SuperHyperVertices such that there's an extreme SuperHyperEdge to have  some extreme SuperHyperVertices uniquely but it isn't an extreme R-SuperHyperMatching. Since it doesn't have  \underline{\textbf{the maximum extreme cardinality}} of an extreme SuperHyperSet $S$ of  extreme SuperHyperVertices such that there's an extreme SuperHyperEdge to have some SuperHyperVertices uniquely. The extreme SuperHyperSet of the extreme SuperHyperVertices $V_{ESHE}\cup \{z\}$ is the maximum extreme cardinality of an extreme SuperHyperSet $S$ of extreme SuperHyperVertices but it isn't an extreme R-SuperHyperMatching. Since it \textbf{\underline{doesn't do}} the extreme procedure such that such that there's an extreme SuperHyperEdge to have some extreme  SuperHyperVertices uniquely  [there are at least one extreme SuperHyperVertex outside  implying there's, sometimes in  the connected extreme SuperHyperGraph $ESHG:(V,E),$ an extreme SuperHyperVertex, titled its extreme SuperHyperNeighbor,  to that extreme SuperHyperVertex in the extreme SuperHyperSet $S$ so as $S$ doesn't do ``the extreme procedure''.]. There's  only \textbf{\underline{one}} extreme SuperHyperVertex    \textbf{\underline{outside}} the intended extreme SuperHyperSet, $V_{ESHE}\cup \{z\},$ in the terms of extreme SuperHyperNeighborhood. Thus the obvious extreme R-SuperHyperMatching,  $V_{ESHE}$ is up. The obvious simple extreme type-SuperHyperSet of the  extreme R-SuperHyperMatching,  $V_{ESHE},$ \textbf{\underline{is}} an extreme SuperHyperSet, $V_{ESHE},$  \textbf{\underline{includes}} only \textbf{\underline{all}}  extreme SuperHyperVertices does forms any kind of extreme pairs are titled   \underline{extreme SuperHyperNeighbors} in a connected extreme SuperHyperGraph $ESHG:(V,E).$ Since the extreme SuperHyperSet of the extreme SuperHyperVertices $V_{ESHE},$ is the \textbf{\underline{maximum extreme SuperHyperCardinality}} of an extreme SuperHyperSet $S$ of  extreme SuperHyperVertices  \textbf{\underline{such that}}  there's an extreme SuperHyperEdge to have some extreme SuperHyperVertices uniquely. Thus, in a connected extreme SuperHyperGraph $ESHG:(V,E).$ Any extreme R-SuperHyperMatching only contains all interior extreme SuperHyperVertices and all exterior extreme SuperHyperVertices from the unique extreme SuperHyperEdge where there's any of them has all possible  extreme SuperHyperNeighbors in and there's all  extreme SuperHyperNeighborhoods in with no exception minus all  extreme SuperHypeNeighbors to some of them not all of them  but everything is possible about extreme SuperHyperNeighborhoods and extreme SuperHyperNeighbors out.
\\
The SuperHyperNotion, namely,  SuperHyperMatching, is up.  There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following extreme SuperHyperSet  of extreme  SuperHyperEdges[SuperHyperVertices] is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching.
 The extreme SuperHyperSet of extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
  is the simple extreme type-SuperHyperSet of the extreme SuperHyperMatching. The extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is an \underline{\textbf{extreme SuperHyperMatching}} $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is an extreme  type-SuperHyperSet with \underline{\textbf{the maximum extreme cardinality}}  of an extreme SuperHyperSet $S$ of extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There are    \underline{not} only \underline{\textbf{two}} extreme SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious  extreme SuperHyperMatching is up. The obvious simple extreme type-SuperHyperSet called the  extreme SuperHyperMatching is an extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} extreme SuperHyperVertices. But the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Doesn't have less than three  SuperHyperVertices \underline{\textbf{inside}} the intended extreme SuperHyperSet. Thus the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching \underline{\textbf{is}} up. To sum them up, the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching. Since the extreme SuperHyperSet of the extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is an  extreme SuperHyperMatching $\mathcal{C}(ESHG)$ for an  extreme SuperHyperGraph $ESHG:(V,E)$ is the extreme SuperHyperSet $S$ of extreme SuperHyperVertices such that there's no an extreme  SuperHyperEdge for some  extreme SuperHyperVertices given by that extreme type-SuperHyperSet  called the  extreme SuperHyperMatching \underline{\textbf{and}} it's an  extreme \underline{\textbf{ SuperHyperMatching}}. Since it\underline{\textbf{'s}}   \underline{\textbf{the maximum extreme cardinality}} of  an extreme SuperHyperSet $S$ of  extreme SuperHyperEdges[SuperHyperVertices] such that there's no  extreme SuperHyperVertex of an extreme SuperHyperEdge is common and there's an   extreme SuperHyperEdge for all extreme SuperHyperVertices. There aren't  only less than three extreme  SuperHyperVertices \underline{\textbf{inside}} the intended  extreme SuperHyperSet,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
 Thus the non-obvious  extreme SuperHyperMatching, \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is up. The obvious simple extreme type-SuperHyperSet of the  extreme SuperHyperMatching, not: \begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Is  the extreme SuperHyperSet, not:
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected extreme  SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple neutrosophic type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``neutrosophic  SuperHyperMatching''}}
\end{center}
 amid those obvious[non-obvious] simple extreme type-SuperHyperSets called the
\begin{center}
 \underline{\textbf{extreme SuperHyperMatching}},
\end{center}
 is only and only
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching}
           \\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
 \mathcal{C}(NSHG)_{Extreme Quasi-SuperHyperMatching SuperHyperPolynomial}
  \\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{2}\rfloor}.
 \\&&
   \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
 \\&&
  \mathcal{C}(NSHG)_{Extreme R-Quasi-SuperHyperMatching SuperHyperPolynomial}=az^{s}+bz^{t}.
\end{eqnarray*}
 In a connected extreme SuperHyperGraph $ESHG:(V,E).$
 \end{proof}
 \begin{example}\label{118EXM23a}
In the extreme Figure \eqref{118NSHG23a}, the connected extreme SuperHyperWheel $NSHW:(V,E),$ is extreme highlighted and featured. The obtained extreme SuperHyperSet, by the Algorithm in previous result, of the extreme SuperHyperVertices of the connected extreme SuperHyperWheel $ESHW:(V,E),$
in the extreme SuperHyperModel \eqref{118NSHG23a}, is the  extreme SuperHyperMatching.

\begin{figure}
\includegraphics[width=100mm]{118NSHG23.png}
 \caption{An extreme SuperHyperWheel extreme Associated to the extreme Notions of  extreme SuperHyperMatching in the extreme Example \eqref{118EXM23a}}
 \label{104NSHG23a}
\end{figure}
 \end{example}
\section{General Extreme Results}
For the SuperHyperMatching,  extreme SuperHyperMatching, and the neutrosophic SuperHyperMatching, some general results are introduced.
\begin{remark}
 Let remind that the neutrosophic SuperHyperMatching is ``redefined'' on the positions of the alphabets.
\end{remark}
\begin{corollary}
 Assume extreme SuperHyperMatching. Then
 \begin{eqnarray*}
&& Neutrosophic ~SuperHyperMatching=\\&&\{the   SuperHyperMatching of the SuperHyperVertices ~|~\\&&\max|SuperHyperOffensive SuperHyper\\&&Clique|_{neutrosophic cardinality amid those SuperHyperMatching.}\}
  \end{eqnarray*}
plus one extreme SuperHypeNeighbor to one. Where $\sigma_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively.

\end{corollary}
\begin{corollary}
Assume a neutrosophic SuperHyperGraph on the same identical letter of the alphabet. Then the notion of neutrosophic SuperHyperMatching and SuperHyperMatching coincide.
\end{corollary}
\begin{corollary}
Assume a neutrosophic SuperHyperGraph on the same identical letter of the alphabet. Then a consecutive sequence of the SuperHyperVertices is a neutrosophic SuperHyperMatching if and only if it's a SuperHyperMatching.
\end{corollary}
\begin{corollary}
Assume a neutrosophic SuperHyperGraph on the same identical letter of the alphabet. Then a consecutive sequence of the SuperHyperVertices is a strongest SuperHyperCycle if and only if it's a longest SuperHyperCycle.
\end{corollary}
\begin{corollary}
Assume SuperHyperClasses of a neutrosophic SuperHyperGraph on the same identical letter of the alphabet. Then its neutrosophic SuperHyperMatching is its SuperHyperMatching and reversely.
\end{corollary}
\begin{corollary}
Assume a neutrosophic SuperHyperPath(-/SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel) on the same identical letter of the alphabet. Then its neutrosophic SuperHyperMatching is its SuperHyperMatching and reversely.
\end{corollary}
\begin{corollary}
 Assume a neutrosophic SuperHyperGraph. Then its neutrosophic SuperHyperMatching isn't well-defined if and only if its SuperHyperMatching isn't well-defined.
\end{corollary}
\begin{corollary}
 Assume SuperHyperClasses of a neutrosophic SuperHyperGraph. Then its neutrosophic SuperHyperMatching isn't well-defined if and only if its SuperHyperMatching isn't well-defined.
\end{corollary}
\begin{corollary}
Assume a neutrosophic SuperHyperPath(-/SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel). Then its neutrosophic SuperHyperMatching isn't well-defined if and only if its SuperHyperMatching isn't well-defined.
\end{corollary}
\begin{corollary}
 Assume a neutrosophic SuperHyperGraph. Then its neutrosophic SuperHyperMatching is well-defined if and only if its SuperHyperMatching is well-defined.
\end{corollary}
\begin{corollary}
 Assume SuperHyperClasses of a neutrosophic SuperHyperGraph. Then its neutrosophic SuperHyperMatching is well-defined if and only if its SuperHyperMatching is well-defined.
\end{corollary}
\begin{corollary}
Assume a neutrosophic SuperHyperPath(-/SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel). Then its neutrosophic SuperHyperMatching is well-defined if and only if its SuperHyperMatching is well-defined.
\end{corollary}

%

\begin{proposition}
Let $ESHG:(V,E)$  be a neutrosophic SuperHyperGraph. Then $V$ is
\begin{itemize}
 \item[$(i):$]  the dual SuperHyperDefensive SuperHyperMatching;
  \item[$(ii):$]  the strong dual SuperHyperDefensive SuperHyperMatching;
    \item[$(iii):$]  the connected dual SuperHyperDefensive SuperHyperMatching;
      \item[$(iv):$]  the $\delta$-dual SuperHyperDefensive SuperHyperMatching;
        \item[$(v):$]  the strong $\delta$-dual SuperHyperDefensive SuperHyperMatching;
          \item[$(vi):$]  the connected $\delta$-dual SuperHyperDefensive SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
Suppose $ESHG:(V,E)$ is a neutrosophic SuperHyperGraph. Consider $V.$ All SuperHyperMembers of $V$ have at least one SuperHyperNeighbor inside the SuperHyperSet more than SuperHyperNeighbor out of SuperHyperSet. Thus,
\\
$(i).$ $V$ is the dual SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in V,~|N(a)\cap V|>|N(a)\cap (V\setminus V)|   \equiv  \\&& \forall a\in V,~|N(a)\cap V|>|N(a)\cap \emptyset|   \equiv \\&&  \forall a\in V,~|N(a)\cap V|>|\emptyset|   \equiv  \\&& \forall a\in V,~|N(a)\cap V|>0   \equiv \\&&  \forall a\in V,~\delta>0.
\end{eqnarray*}
$(ii).$ $V$ is the strong dual SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N_s(a)\cap S|>|N_s(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in V,~|N_s(a)\cap V|>|N_s(a)\cap (V\setminus V)|   \equiv  \\&& \forall a\in V,~|N_s(a)\cap V|>|N_s(a)\cap \emptyset|   \equiv \\&&  \forall a\in V,~|N_s(a)\cap V|>|\emptyset|   \equiv  \\&& \forall a\in V,~|N_s(a)\cap V|>0   \equiv \\&&  \forall a\in V,~\delta>0.\end{eqnarray*}
$(iii).$ $V$ is the connected dual SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N_c(a)\cap S|>|N_c(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in V,~|N_c(a)\cap V|>|N_c(a)\cap (V\setminus V)|   \equiv  \\&& \forall a\in V,~|N_c(a)\cap V|>|N_c(a)\cap \emptyset|   \equiv \\&&  \forall a\in V,~|N_c(a)\cap V|>|\emptyset|   \equiv  \\&& \forall a\in V,~|N_c(a)\cap V|>0   \equiv \\&&  \forall a\in V,~\delta>0.
\end{eqnarray*}
$(iv).$ $V$ is the $\delta$-dual SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N(a)\cap S)-(N(a)\cap (V\setminus S))| > \delta  \equiv \\&&
 \forall a\in V,~|(N(a)\cap V)-(N(a)\cap (V\setminus V))| > \delta  \equiv \\&&
 \forall a\in V,~|(N(a)\cap V)-(N(a)\cap (\emptyset))| > \delta  \equiv \\&&
 \forall a\in V,~|(N(a)\cap V)-(\emptyset)| > \delta  \equiv \\&&
 \forall a\in V,~|(N(a)\cap V)| > \delta.
\end{eqnarray*}
$(v).$ $V$ is the strong $\delta$-dual SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N_s(a)\cap S)-(N_s(a)\cap (V\setminus S))| > \delta  \equiv \\&&
 \forall a\in V,~|(N_s(a)\cap V)-(N_s(a)\cap (V\setminus V))| > \delta  \equiv \\&&
 \forall a\in V,~|(N_s(a)\cap V)-(N_s(a)\cap (\emptyset))| > \delta  \equiv \\&&
 \forall a\in V,~|(N_s(a)\cap V)-(\emptyset)| > \delta  \equiv \\&&
 \forall a\in V,~|(N_s(a)\cap V)| > \delta.
\end{eqnarray*}
$(vi).$ $V$ is connected $\delta$-dual SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N_c(a)\cap S)-(N_c(a)\cap (V\setminus S))| > \delta  \equiv \\&&
 \forall a\in V,~|(N_c(a)\cap V)-(N_c(a)\cap (V\setminus V))| > \delta  \equiv \\&&
 \forall a\in V,~|(N_c(a)\cap V)-(N_c(a)\cap (\emptyset))| > \delta  \equiv \\&&
 \forall a\in V,~|(N_c(a)\cap V)-(\emptyset)| > \delta  \equiv \\&&
 \forall a\in V,~|(N_c(a)\cap V)| > \delta.
\end{eqnarray*}
\end{proof}
\begin{proposition}
Let $NTG:(V,E,\sigma,\mu)$  be a neutrosophic SuperHyperGraph. Then $\emptyset$ is
\begin{itemize}
 \item[$(i):$] the SuperHyperDefensive SuperHyperMatching;
  \item[$(ii):$]  the strong SuperHyperDefensive SuperHyperMatching;
    \item[$(iii):$]  the connected defensive SuperHyperDefensive SuperHyperMatching;
      \item[$(iv):$]  the $\delta$-SuperHyperDefensive SuperHyperMatching;
        \item[$(v):$]  the strong $\delta$-SuperHyperDefensive SuperHyperMatching;
          \item[$(vi):$]  the connected $\delta$-SuperHyperDefensive SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
Suppose $ESHG:(V,E)$ is a neutrosophic SuperHyperGraph. Consider $\emptyset.$ All SuperHyperMembers of $\emptyset$ have no SuperHyperNeighbor inside the SuperHyperSet less than SuperHyperNeighbor out of SuperHyperSet. Thus,
\\
$(i).$ $\emptyset$ is the SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in \emptyset,~|N(a)\cap \emptyset|<|N(a)\cap (V\setminus \emptyset)|   \equiv  \\&& \forall a\in\emptyset,~|\emptyset|<|N(a)\cap (V\setminus \emptyset)|   \equiv \\&&  \forall a\in \emptyset,~0<|N(a)\cap V|   \equiv  \\&& \forall a\in \emptyset,~0<|N(a)\cap V|   \equiv \\&&  \forall a\in V,~\delta>0.
\end{eqnarray*}
$(ii).$ $\emptyset$ is the strong SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N_s(a)\cap S|<|N_s(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in \emptyset,~|N_s(a)\cap \emptyset|<|N_s(a)\cap (V\setminus \emptyset)|   \equiv  \\&& \forall a\in\emptyset,~|\emptyset|<|N_s(a)\cap (V\setminus \emptyset)|   \equiv \\&&  \forall a\in \emptyset,~0<|N_s(a)\cap V|   \equiv  \\&& \forall a\in \emptyset,~0<|N_s(a)\cap V|   \equiv \\&&  \forall a\in V,~\delta>0.
\end{eqnarray*}
$(iii).$ $\emptyset$ is the connected SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N_c(a)\cap S|<|N_c(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in \emptyset,~|N_c(a)\cap \emptyset|<|N_c(a)\cap (V\setminus \emptyset)|   \equiv  \\&& \forall a\in\emptyset,~|\emptyset|<|N_c(a)\cap (V\setminus \emptyset)|   \equiv \\&&  \forall a\in \emptyset,~0<|N_c(a)\cap V|   \equiv  \\&& \forall a\in \emptyset,~0<|N_c(a)\cap V|   \equiv \\&&  \forall a\in V,~\delta>0.
\end{eqnarray*}
$(iv).$ $\emptyset$ is the $\delta$-SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N(a)\cap S)-(N(a)\cap (V\setminus S))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|(N(a)\cap \emptyset)-(N(a)\cap (V\setminus \emptyset))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|(N(a)\cap \emptyset)-(N(a)\cap (V))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|\emptyset| < \delta  \equiv \\&&
 \forall a\in V,~0 < \delta.
\end{eqnarray*}
$(v).$ $\emptyset$ is the strong $\delta$-SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N_s(a)\cap S)-(N_s(a)\cap (V\setminus S))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|(N_s(a)\cap \emptyset)-(N_s(a)\cap (V\setminus \emptyset))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|(N_s(a)\cap \emptyset)-(N_s(a)\cap (V))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|\emptyset| < \delta  \equiv \\&&
 \forall a\in V,~0 < \delta.
\end{eqnarray*}
$(vi).$ $\emptyset$ is the connected $\delta$-SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N_c(a)\cap S)-(N_c(a)\cap (V\setminus S))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|(N_c(a)\cap \emptyset)-(N_c(a)\cap (V\setminus \emptyset))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|(N_c(a)\cap \emptyset)-(N_c(a)\cap (V))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|\emptyset| < \delta  \equiv \\&&
 \forall a\in V,~0 < \delta.
\end{eqnarray*}
\end{proof}
\begin{proposition}
Let $ESHG:(V,E)$ be a neutrosophic SuperHyperGraph. Then an independent SuperHyperSet is
\begin{itemize}
 \item[$(i):$] the SuperHyperDefensive SuperHyperMatching;
  \item[$(ii):$]  the strong SuperHyperDefensive SuperHyperMatching;
    \item[$(iii):$]  the connected SuperHyperDefensive SuperHyperMatching;
      \item[$(iv):$]  the $\delta$-SuperHyperDefensive SuperHyperMatching;
        \item[$(v):$]  the strong $\delta$-SuperHyperDefensive SuperHyperMatching;
          \item[$(vi):$]  the connected $\delta$-SuperHyperDefensive SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
Suppose $ESHG:(V,E)$ is a neutrosophic SuperHyperGraph. Consider $S.$ All SuperHyperMembers of $S$ have no SuperHyperNeighbor inside the SuperHyperSet less than SuperHyperNeighbor out of SuperHyperSet. Thus,
\\
$(i).$ An independent SuperHyperSet is the SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|   \equiv  \\&& \forall a\in S,~|\emptyset|<|N(a)\cap (V\setminus S)|   \equiv \\&&  \forall a\in S,~0<|N(a)\cap V|   \equiv  \\&& \forall a\in S,~0<|N(a)|   \equiv \\&&  \forall a\in V,~\delta>0.
\end{eqnarray*}
$(ii).$ An independent SuperHyperSet is the strong SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N_s(a)\cap S|<|N_s(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in S,~|N_s(a)\cap S|<|N_s(a)\cap (V\setminus S)|   \equiv  \\&& \forall a\in S,~|\emptyset|<|N_s(a)\cap (V\setminus S)|   \equiv \\&&  \forall a\in S,~0<|N_s(a)\cap V|   \equiv  \\&& \forall a\in S,~0<|N_s(a)|   \equiv \\&&  \forall a\in V,~\delta>0.
\end{eqnarray*}
$(iii).$ An independent SuperHyperSet is the connected SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N_c(a)\cap S|<|N_c(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in S,~|N_c(a)\cap S|<|N_c(a)\cap (V\setminus S)|   \equiv  \\&& \forall a\in S,~|\emptyset|<|N_c(a)\cap (V\setminus S)|   \equiv \\&&  \forall a\in S,~0<|N_c(a)\cap V|   \equiv  \\&& \forall a\in S,~0<|N_c(a)|   \equiv \\&&  \forall a\in V,~\delta>0.
\end{eqnarray*}
$(iv).$ An independent SuperHyperSet is the $\delta$-SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N(a)\cap S)-(N(a)\cap (V\setminus S))| < \delta  \equiv \\&&
 \forall a\in S,~|(N(a)\cap S)-(N(a)\cap (V\setminus S))| < \delta  \equiv \\&&
 \forall a\in S,~|(N(a)\cap S)-(N(a)\cap (V))| < \delta  \equiv \\&&
 \forall a\in S,~|\emptyset| < \delta  \equiv \\&&
 \forall a\in V,~0 < \delta.
\end{eqnarray*}
$(v).$ An independent SuperHyperSet is the strong $\delta$-SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N_s(a)\cap S)-(N_s(a)\cap (V\setminus S))| < \delta  \equiv \\&&
 \forall a\in S,~|(N_s(a)\cap S)-(N_s(a)\cap (V\setminus S))| < \delta  \equiv \\&&
 \forall a\in S,~|(N_s(a)\cap S)-(N_s(a)\cap (V))| < \delta  \equiv \\&&
 \forall a\in S,~|\emptyset| < \delta  \equiv \\&&
 \forall a\in V,~0 < \delta.
\end{eqnarray*}
$(vi).$ An independent SuperHyperSet is the connected $\delta$-SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N_c(a)\cap S)-(N_c(a)\cap (V\setminus S))| < \delta  \equiv \\&&
 \forall a\in S,~|(N_c(a)\cap S)-(N_c(a)\cap (V\setminus S))| < \delta  \equiv \\&&
 \forall a\in S,~|(N_c(a)\cap S)-(N_c(a)\cap (V))| < \delta  \equiv \\&&
 \forall a\in S,~|\emptyset| < \delta  \equiv \\&&
 \forall a\in V,~0 < \delta.
\end{eqnarray*}
\end{proof}

\begin{proposition}
Let $ESHG:(V,E)$ be a neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperCycle/SuperHyperPath. Then $V$ is a maximal
\begin{itemize}
 \item[$(i):$] SuperHyperDefensive SuperHyperMatching;
  \item[$(ii):$]  strong SuperHyperDefensive SuperHyperMatching;
    \item[$(iii):$]  connected SuperHyperDefensive SuperHyperMatching;
      \item[$(iv):$]  $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyperMatching;
        \item[$(v):$]  strong $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyperMatching;
          \item[$(vi):$]  connected $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyperMatching;
\end{itemize}
Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proof}
Suppose $ESHG:(V,E)$ is a neutrosophic SuperHyperGraph which is a SuperHyperUniform SuperHyperCycle/SuperHyperPath.\\
$(i).$
Consider one segment is out of $S$ which is SuperHyperDefensive SuperHyperMatching. This segment has $2t$ SuperHyperNeighbors in $S$, i.e, Suppose $x_{i_{i=1,2,\ldots,t}}\in V\setminus S$ such that $y_{i_{i=1,2,\ldots,t}},z_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}}).$ By it's the exterior SuperHyperVertices and the interior SuperHyperVertices coincide and it's SuperHyperUniform SuperHyperCycle, $|N(x_{i_{i=1,2,\ldots,t}})|=|N(y_{i_{i=1,2,\ldots,t}})|=|N(z_{i_{i=1,2,\ldots,t}})|=2t.$ Thus
 \begin{eqnarray*}&&
 \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|   \equiv  \\&& \exists y_{i_{i=1,2,\ldots,t}}\in V\setminus \{x_i\}_{i=1}^{t},~|N(y_{i_{i=1,2,\ldots,t}})\cap S|< \\&&|N(y_{i_{i=1,2,\ldots,t}})\cap (V\setminus (V\setminus \{x_{i_{i=1,2,\ldots,t}}\}))|   \equiv \\&&  \exists y_{i_{i=1,2,\ldots,t}}\in V\setminus \{x_i\}_{i=1}^{t},~|N(y_{i_{i=1,2,\ldots,t}})\cap S|< \\&&|N(y_{i_{i=1,2,\ldots,t}})\cap \{x_{i_{i=1,2,\ldots,t}}\})|   \equiv \\&&  \exists y_{i_{i=1,2,\ldots,t}}\in V\setminus \{x_i\}_{i=1}^{t},~|\{z_1,z_2,\ldots,z_{t-1}\}|< \\&&|\{x_1,x_2,\ldots,x_{t-1}\})|   \equiv \\&&  \exists y\in S,~t-1<t-1.
\end{eqnarray*}
Thus it's contradiction. It implies every $V\setminus\{x_{i_{i=1,2,\ldots,t}}\}$ isn't SuperHyperDefensive SuperHyperMatching in a given SuperHyperUniform SuperHyperCycle.
\\
Consider one segment, with two segments related to the SuperHyperLeaves as exceptions, is out of $S$ which is SuperHyperDefensive SuperHyperMatching. This segment has $2t$ SuperHyperNeighbors in $S$, i.e, Suppose $x_{i_{i=1,2,\ldots,t}}\in V\setminus S$ such that $y_{i_{i=1,2,\ldots,t}},z_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}}).$ By it's the exterior SuperHyperVertices and the interior SuperHyperVertices coincide and it's SuperHyperUniform SuperHyperPath, $|N(x_{i_{i=1,2,\ldots,t}})|=|N(y_{i_{i=1,2,\ldots,t}})|=|N(z_{i_{i=1,2,\ldots,t}})|=2t.$ Thus
 \begin{eqnarray*}&&
 \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|   \equiv  \\&& \exists y_{i_{i=1,2,\ldots,t}}\in V\setminus \{x_i\}_{i=1}^{t},~|N(y_{i_{i=1,2,\ldots,t}})\cap S|< \\&&|N(y_{i_{i=1,2,\ldots,t}})\cap (V\setminus (V\setminus \{x_{i_{i=1,2,\ldots,t}}\}))|   \equiv \\&&  \exists y_{i_{i=1,2,\ldots,t}}\in V\setminus \{x_i\}_{i=1}^{t},~|N(y_{i_{i=1,2,\ldots,t}})\cap S|< \\&&|N(y_{i_{i=1,2,\ldots,t}})\cap \{x_{i_{i=1,2,\ldots,t}}\})|   \equiv \\&&  \exists y_{i_{i=1,2,\ldots,t}}\in V\setminus \{x_i\}_{i=1}^{t},~|\{z_1,z_2,\ldots,z_{t-1}\}|< \\&&|\{x_1,x_2,\ldots,x_{t-1}\})|   \equiv \\&&  \exists y\in S,~t-1<t-1.
\end{eqnarray*}
Thus it's contradiction. It implies every $V\setminus\{x_{i_{i=1,2,\ldots,t}}\}$ isn't SuperHyperDefensive SuperHyperMatching in a given SuperHyperUniform SuperHyperPath.
\\
$(ii),$ $(iii)$ are obvious by $(i).$
\\
$(iv).$ By $(i),$ $|V|$ is maximal and it's a SuperHyperDefensive SuperHyperMatching. Thus it's $|V|$-SuperHyperDefensive SuperHyperMatching.
\\
$(v),(vi)$ are obvious by $(iv).$
\end{proof}
\begin{proposition}
Let $ESHG:(V,E)$ be a neutrosophic SuperHyperGraph which is a  SuperHyperUniform SuperHyperWheel. Then $V$ is a maximal
\begin{itemize}
 \item[$(i):$] dual SuperHyperDefensive SuperHyperMatching;
  \item[$(ii):$]  strong dual SuperHyperDefensive SuperHyperMatching;
    \item[$(iii):$]  connected dual SuperHyperDefensive SuperHyperMatching;
      \item[$(iv):$]  $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyperMatching;
        \item[$(v):$]  strong $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyperMatching;
          \item[$(vi):$]  connected $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyperMatching;
\end{itemize}
Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proof}
Suppose $ESHG:(V,E)$ is a neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperWheel.\\
$(i).$ Consider one segment is out of $S$ which is SuperHyperDefensive SuperHyperMatching. This segment has $3t$ SuperHyperNeighbors in $S$, i.e, Suppose $x_{i_{i=1,2,\ldots,t}}\in V\setminus S$ such that $y_{i_{i=1,2,\ldots,t}},z_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}}).$ By it's the exterior SuperHyperVertices and the interior SuperHyperVertices coincide and it's SuperHyperUniform SuperHyperWheel, $|N(x_{i_{i=1,2,\ldots,t}})|=|N(y_{i_{i=1,2,\ldots,t}})|=|N(z_{i_{i=1,2,\ldots,t}})|=3t.$ Thus
 \begin{eqnarray*}&&
 \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|   \equiv  \\&& \exists y_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}})\in V\setminus \{x_i\}_{i=1}^{t},\\&&~|N(y_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}}))\cap S|<\\&&|N(y_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}}))\cap (V\setminus (V\setminus \{x_{i_{i=1,2,\ldots,t}}\}))|   \equiv \\&&  \exists y_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}})\in V\setminus \{x_i\}_{i=1}^{t},\\&&~|N(y_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}}))\cap S|<\\&&|N(y_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}}))\cap \{x_{i_{i=1,2,\ldots,t}}\})|   \equiv \\&&  \exists y_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}})\in V\setminus \{x_i\}_{i=1}^{t},\\&&~|\{z_1,z_2,\ldots,z_{t-1},z'_1,z'_2,\ldots,z'_t\}|<|\{x_1,x_2,\ldots,x_{t-1}\})|   \equiv \\&&  \exists y\in S,~2t-1<t-1.
\end{eqnarray*}
Thus it's contradiction. It implies every $V\setminus\{x_{i_{i=1,2,\ldots,t}}\}$ is SuperHyperDefensive SuperHyperMatching in a given SuperHyperUniform SuperHyperWheel.
\\
$(ii),$ $(iii)$ are obvious by $(i).$
\\
$(iv).$ By $(i),$ $|V|$ is maximal and it is a dual SuperHyperDefensive SuperHyperMatching. Thus it's a dual $|V|$-SuperHyperDefensive SuperHyperMatching.
\\
$(v),(vi)$ are obvious by $(iv).$
\end{proof}

\begin{proposition}
Let $ESHG:(V,E)$ be a neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperCycle/SuperHyperPath. Then the number of
\begin{itemize}
 \item[$(i):$] the SuperHyperMatching;
  \item[$(ii):$] the SuperHyperMatching;
    \item[$(iii):$]  the connected SuperHyperMatching;
      \item[$(iv):$]  the $\mathcal{O}(ESHG)$-SuperHyperMatching;
        \item[$(v):$]  the strong $\mathcal{O}(ESHG)$-SuperHyperMatching;
          \item[$(vi):$]  the connected $\mathcal{O}(ESHG)$-SuperHyperMatching.
\end{itemize}
is one and it's only $V.$ Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proof}
Suppose $ESHG:(V,E)$ is a neutrosophic SuperHyperGraph which is a SuperHyperUniform SuperHyperCycle/SuperHyperPath.\\
$(i).$
Consider one segment is out of $S$ which is SuperHyperDefensive SuperHyperMatching. This segment has $2t$ SuperHyperNeighbors in $S$, i.e, Suppose $x_{i_{i=1,2,\ldots,t}}\in V\setminus S$ such that $y_{i_{i=1,2,\ldots,t}},z_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}}).$ By it's the exterior SuperHyperVertices and the interior SuperHyperVertices coincide and it's SuperHyperUniform SuperHyperCycle, $|N(x_{i_{i=1,2,\ldots,t}})|=|N(y_{i_{i=1,2,\ldots,t}})|=|N(z_{i_{i=1,2,\ldots,t}})|=2t.$ Thus
 \begin{eqnarray*}&&
 \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|   \equiv  \\&& \exists y_{i_{i=1,2,\ldots,t}}\in V\setminus \{x_i\}_{i=1}^{t},~|N(y_{i_{i=1,2,\ldots,t}})\cap S|<\\&&|N(y_{i_{i=1,2,\ldots,t}})\cap (V\setminus (V\setminus \{x_{i_{i=1,2,\ldots,t}}\}))|   \equiv \\&&  \exists y_{i_{i=1,2,\ldots,t}}\in V\setminus \{x_i\}_{i=1}^{t},~|N(y_{i_{i=1,2,\ldots,t}})\cap S|<\\&&|N(y_{i_{i=1,2,\ldots,t}})\cap \{x_{i_{i=1,2,\ldots,t}}\})|   \equiv \\&&  \exists y_{i_{i=1,2,\ldots,t}}\in V\setminus \{x_i\}_{i=1}^{t},~|\{z_1,z_2,\ldots,z_{t-1}\}|<|\{x_1,x_2,\ldots,x_{t-1}\})|   \equiv \\&&  \exists y\in S,~t-1<t-1.
\end{eqnarray*}
Thus it's contradiction. It implies every $V\setminus\{x_{i_{i=1,2,\ldots,t}}\}$ isn't SuperHyperDefensive SuperHyperMatching in a given SuperHyperUniform SuperHyperCycle.
\\
Consider one segment, with two segments related to the SuperHyperLeaves as exceptions, is out of $S$ which is SuperHyperDefensive SuperHyperMatching. This segment has $2t$ SuperHyperNeighbors in $S$, i.e, Suppose $x_{i_{i=1,2,\ldots,t}}\in V\setminus S$ such that $y_{i_{i=1,2,\ldots,t}},z_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}}).$ By it's the exterior SuperHyperVertices and the interior SuperHyperVertices coincide and it's SuperHyperUniform SuperHyperPath, $|N(x_{i_{i=1,2,\ldots,t}})|=|N(y_{i_{i=1,2,\ldots,t}})|=|N(z_{i_{i=1,2,\ldots,t}})|=2t.$ Thus
 \begin{eqnarray*}&&
 \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|   \equiv  \\&& \exists y_{i_{i=1,2,\ldots,t}}\in V\setminus \{x_i\}_{i=1}^{t},~|N(y_{i_{i=1,2,\ldots,t}})\cap S|<\\&&|N(y_{i_{i=1,2,\ldots,t}})\cap (V\setminus (V\setminus \{x_{i_{i=1,2,\ldots,t}}\}))|   \equiv \\&&  \exists y_{i_{i=1,2,\ldots,t}}\in V\setminus \{x_i\}_{i=1}^{t},~|N(y_{i_{i=1,2,\ldots,t}})\cap S|<\\&&|N(y_{i_{i=1,2,\ldots,t}})\cap \{x_{i_{i=1,2,\ldots,t}}\})|   \equiv \\&&  \exists y_{i_{i=1,2,\ldots,t}}\in V\setminus \{x_i\}_{i=1}^{t},~|\{z_1,z_2,\ldots,z_{t-1}\}|<\\&&|\{x_1,x_2,\ldots,x_{t-1}\})|   \equiv \\&&  \exists y\in S,~t-1<t-1.
\end{eqnarray*}
Thus it's contradiction. It implies every $V\setminus\{x_{i_{i=1,2,\ldots,t}}\}$ isn't SuperHyperDefensive SuperHyperMatching in a given SuperHyperUniform SuperHyperPath.
\\
$(ii),$ $(iii)$ are obvious by $(i).$
\\
$(iv).$ By $(i),$ $|V|$ is maximal and it's a SuperHyperDefensive SuperHyperMatching. Thus it's $|V|$-SuperHyperDefensive SuperHyperMatching.
\\
$(v),(vi)$ are obvious by $(iv).$
\end{proof}

\begin{proposition}
Let $ESHG:(V,E)$ be a neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperWheel. Then the number of
\begin{itemize}
 \item[$(i):$] the dual SuperHyperMatching;
  \item[$(ii):$] the dual  SuperHyperMatching;
    \item[$(iii):$]  the dual connected SuperHyperMatching;
      \item[$(iv):$]  the dual $\mathcal{O}(ESHG)$-SuperHyperMatching;
        \item[$(v):$]  the strong dual $\mathcal{O}(ESHG)$-SuperHyperMatching;
          \item[$(vi):$]  the connected dual $\mathcal{O}(ESHG)$-SuperHyperMatching.
\end{itemize}
is one and it's only $V.$ Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proof}
Suppose $ESHG:(V,E)$ is a neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperWheel.\\
$(i).$ Consider one segment is out of $S$ which is SuperHyperDefensive SuperHyperMatching. This segment has $3t$ SuperHyperNeighbors in $S$, i.e, Suppose $x_{i_{i=1,2,\ldots,t}}\in V\setminus S$ such that $y_{i_{i=1,2,\ldots,t}},z_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}}).$ By it's the exterior SuperHyperVertices and the interior SuperHyperVertices coincide and it's SuperHyperUniform SuperHyperWheel, $|N(x_{i_{i=1,2,\ldots,t}})|=|N(y_{i_{i=1,2,\ldots,t}})|=|N(z_{i_{i=1,2,\ldots,t}})|=3t.$ Thus
 \begin{eqnarray*}&&
 \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|   \equiv  \\&& \exists y_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}})\in V\setminus \{x_i\}_{i=1}^{t},\\&&~|N(y_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}}))\cap S|<\\&&|N(y_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}}))\cap (V\setminus (V\setminus \{x_{i_{i=1,2,\ldots,t}}\}))|   \equiv \\&&  \exists y_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}})\in V\setminus \{x_i\}_{i=1}^{t}\\&&,~|N(y_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}}))\cap S|<\\&&|N(y_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}}))\cap \{x_{i_{i=1,2,\ldots,t}}\})|   \equiv \\&&  \exists y_{i_{i=1,2,\ldots,t}},s_{i_{i=1,2,\ldots,t}}\in N(x_{i_{i=1,2,\ldots,t}})\in V\setminus \{x_i\}_{i=1}^{t},\\&&~|\{z_1,z_2,\ldots,z_{t-1},z'_1,z'_2,\ldots,z'_t\}|<|\{x_1,x_2,\ldots,x_{t-1}\})|   \equiv \\&&  \exists y\in S,~2t-1<t-1.
\end{eqnarray*}
Thus it's contradiction. It implies every $V\setminus\{x_{i_{i=1,2,\ldots,t}}\}$ isn't a dual SuperHyperDefensive SuperHyperMatching in a given SuperHyperUniform SuperHyperWheel.
\\
$(ii),$ $(iii)$ are obvious by $(i).$
\\
$(iv).$ By $(i),$ $|V|$ is maximal and it's a dual SuperHyperDefensive SuperHyperMatching. Thus it isn't an $|V|$-SuperHyperDefensive SuperHyperMatching.
\\
$(v),(vi)$ are obvious by $(iv).$
\end{proof}
\begin{proposition}
Let $ESHG:(V,E)$ be a neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperStar/SuperHyperComplete SuperHyperBipartite/SuperHyperComplete SuperHyperMultipartite. Then a SuperHyperSet contains [the SuperHyperCenter and] the half of multiplying $r$ with the number of all the SuperHyperEdges plus one of all the SuperHyperVertices is a
\begin{itemize}
 \item[$(i):$] dual SuperHyperDefensive SuperHyperMatching;
  \item[$(ii):$]  strong dual SuperHyperDefensive SuperHyperMatching;
    \item[$(iii):$]  connected dual SuperHyperDefensive SuperHyperMatching;
      \item[$(iv):$]  $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperMatching;
        \item[$(v):$]  strong $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperMatching;
          \item[$(vi):$]  connected $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$
 Consider $n$ half $+1$ SuperHyperVertices are in $S$ which is SuperHyperDefensive SuperHyperMatching. A SuperHyperVertex has either $\frac{n}{2}$ or one  SuperHyperNeighbors in $S.$ If the SuperHyperVertex is non-SuperHyperCenter, then
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~1>0.
 \end{eqnarray*}
If the SuperHyperVertex is SuperHyperCenter, then
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{n}{2}>\frac{n}{2}-1.
 \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a dual SuperHyperDefensive SuperHyperMatching in a given SuperHyperStar.
 \\
Consider $n$ half $+1$ SuperHyperVertices are in $S$ which is SuperHyperDefensive SuperHyperMatching. A SuperHyperVertex has at most $\frac{n}{2}$ SuperHyperNeighbors in $S.$
 \begin{eqnarray*}&&
  \forall a\in S,~\frac{n}{2}>|N(a)\cap S|>\frac{n}{2}-1>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{n}{2}>\frac{n}{2}-1.
 \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a dual SuperHyperDefensive SuperHyperMatching in a given SuperHyperComplete SuperHyperBipartite which isn't a SuperHyperStar.
 \\
 Consider $n$ half $+1$ SuperHyperVertices are in $S$ which is SuperHyperDefensive SuperHyperMatching and they're chosen from different SuperHyperParts, equally or almost equally as possible. A SuperHyperVertex has at most $\frac{n}{2}$ SuperHyperNeighbors in $S.$
 \begin{eqnarray*}&&
  \forall a\in S,~\frac{n}{2}>|N(a)\cap S|>\frac{n}{2}-1>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{n}{2}>\frac{n}{2}-1.
 \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a dual SuperHyperDefensive SuperHyperMatching in a given SuperHyperComplete SuperHyperMultipartite which is neither a SuperHyperStar nor SuperHyperComplete SuperHyperBipartite.
\\
$(ii),$ $(iii)$ are obvious by $(i).$
\\
$(iv).$ By $(i),$ $\{x_i\}_{i=1}^{\frac{\mathcal{O}(ESHG)}{2}+1}$ is a dual SuperHyperDefensive SuperHyperMatching. Thus it's $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperMatching.
\\
$(v),(vi)$ are obvious by $(iv).$
\end{proof}
\begin{proposition}
Let $ESHG:(V,E)$ be a neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperStar/SuperHyperComplete SuperHyperBipartite/SuperHyperComplete SuperHyperMultipartite. Then a SuperHyperSet contains the half of multiplying $r$ with the number of all the SuperHyperEdges plus one of all the SuperHyperVertices in the biggest SuperHyperPart is a
\begin{itemize}
 \item[$(i):$]
 SuperHyperDefensive SuperHyperMatching;
  \item[$(ii):$]  strong  SuperHyperDefensive SuperHyperMatching;
    \item[$(iii):$]  connected  SuperHyperDefensive SuperHyperMatching;
      \item[$(iv):$]  $\delta$-SuperHyperDefensive SuperHyperMatching;
        \item[$(v):$]  strong $\delta$-SuperHyperDefensive SuperHyperMatching;
          \item[$(vi):$]  connected $\delta$-SuperHyperDefensive SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$
 Consider the half of multiplying $r$ with the number of all the SuperHyperEdges plus one of all the SuperHyperVertices in the biggest SuperHyperPart are in $S$ which is SuperHyperDefensive SuperHyperMatching. A SuperHyperVertex has either $n-1,1$ or zero  SuperHyperNeighbors in $S.$ If the SuperHyperVertex is in $S,$ then
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~0<1.
 \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a SuperHyperDefensive SuperHyperMatching in a given SuperHyperStar.
 \\
Consider the half of multiplying $r$ with the number of all the SuperHyperEdges plus one of all the SuperHyperVertices in the biggest SuperHyperPart  are in $S$ which is SuperHyperDefensive SuperHyperMatching. A SuperHyperVertex has no SuperHyperNeighbor in $S.$
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~0<\delta.
 \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a SuperHyperDefensive SuperHyperMatching in a given SuperHyperComplete SuperHyperBipartite which isn't a SuperHyperStar.
 \\
 Consider the half of multiplying $r$ with the number of all the SuperHyperEdges plus one of all the SuperHyperVertices in the biggest SuperHyperPart  are in $S$ which is SuperHyperDefensive SuperHyperMatching. A SuperHyperVertex has no SuperHyperNeighbor in $S.$
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~0<\delta.
 \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a SuperHyperDefensive SuperHyperMatching in a given SuperHyperComplete SuperHyperMultipartite which is neither a SuperHyperStar nor SuperHyperComplete SuperHyperBipartite.
\\
$(ii),$ $(iii)$ are obvious by $(i).$
\\
$(iv).$ By $(i),$ $S$ is a SuperHyperDefensive SuperHyperMatching. Thus it's an $\delta$-SuperHyperDefensive SuperHyperMatching.
\\
$(v),(vi)$ are obvious by $(iv).$
\end{proof}

\begin{proposition}
Let $ESHG:(V,E)$ be a neutrosophic SuperHyperUniform SuperHyperGraph which is a SuperHyperStar/SuperHyperComplete SuperHyperBipartite/SuperHyperComplete SuperHyperMultipartite. Then Then the number of \begin{itemize}
 \item[$(i):$] dual SuperHyperDefensive SuperHyperMatching;
  \item[$(ii):$]  strong dual SuperHyperDefensive SuperHyperMatching;
    \item[$(iii):$]  connected dual SuperHyperDefensive SuperHyperMatching;
      \item[$(iv):$]  $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperMatching;
        \item[$(v):$]  strong $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperMatching;
          \item[$(vi):$]  connected $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperMatching.
\end{itemize}
is one and it's only $S,$ a SuperHyperSet contains [the SuperHyperCenter and] the half of multiplying $r$ with the number of all the SuperHyperEdges plus one of all the SuperHyperVertices. Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proof}
 $(i).$
 Consider $n$ half $+1$ SuperHyperVertices are in $S$ which is SuperHyperDefensive SuperHyperMatching. A SuperHyperVertex has either $\frac{n}{2}$ or one  SuperHyperNeighbors in $S.$ If the SuperHyperVertex is non-SuperHyperCenter, then
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~1>0.
 \end{eqnarray*}
If the SuperHyperVertex is SuperHyperCenter, then
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{n}{2}>\frac{n}{2}-1.  
 \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a dual SuperHyperDefensive SuperHyperMatching in a given SuperHyperStar.
 \\
Consider $n$ half $+1$ SuperHyperVertices are in $S$ which is SuperHyperDefensive SuperHyperMatching. A SuperHyperVertex has at most $\frac{n}{2}$ SuperHyperNeighbors in $S.$
 \begin{eqnarray*}&&
  \forall a\in S,~\frac{n}{2}>|N(a)\cap S|>\frac{n}{2}-1>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{n}{2}>\frac{n}{2}-1.
 \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a dual SuperHyperDefensive SuperHyperMatching in a given SuperHyperComplete SuperHyperBipartite which isn't a SuperHyperStar.
 \\
 Consider $n$ half $+1$ SuperHyperVertices are in $S$ which is SuperHyperDefensive SuperHyperMatching and they're chosen from different SuperHyperParts, equally or almost equally as possible. A SuperHyperVertex has at most $\frac{n}{2}$ SuperHyperNeighbors in $S.$
 \begin{eqnarray*}&&
  \forall a\in S,~\frac{n}{2}>|N(a)\cap S|>\frac{n}{2}-1>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{n}{2}>\frac{n}{2}-1.  
 \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a dual SuperHyperDefensive SuperHyperMatching in a given SuperHyperComplete SuperHyperMultipartite which is neither a SuperHyperStar nor SuperHyperComplete SuperHyperBipartite.
\\
$(ii),$ $(iii)$ are obvious by $(i).$
\\
$(iv).$ By $(i),$ $\{x_i\}_{i=1}^{\frac{\mathcal{O}(ESHG)}{2}+1}$ is a dual SuperHyperDefensive SuperHyperMatching. Thus it's $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperMatching.
\\
$(v),(vi)$ are obvious by $(iv).$
\end{proof}

\begin{proposition}
Let $ESHG:(V,E)$  be a neutrosophic SuperHyperGraph. The number of connected component is $|V-S|$ if there's a SuperHyperSet which is a dual
\begin{itemize}
 \item[$(i):$]  SuperHyperDefensive SuperHyperMatching;
  \item[$(ii):$]  strong SuperHyperDefensive SuperHyperMatching;
    \item[$(iii):$] connected SuperHyperDefensive SuperHyperMatching;
      \item[$(iv):$]  SuperHyperMatching;
        \item[$(v):$]  strong 1-SuperHyperDefensive SuperHyperMatching;
          \item[$(vi):$]  connected 1-SuperHyperDefensive SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$
 Consider some SuperHyperVertices are out of $S$ which is a dual SuperHyperDefensive SuperHyperMatching. These SuperHyperVertex-type have some SuperHyperNeighbors in $S $ but no SuperHyperNeighbor out of $S.$ Thus
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~1>0.
 \end{eqnarray*}
Thus it's proved. It implies every $S$ is a dual SuperHyperDefensive SuperHyperMatching and number of connected component is $|V-S|.$
 \\
$(ii),$ $(iii)$ are obvious by $(i).$
\\
$(iv).$ By $(i),$ $S$ is a dual SuperHyperDefensive SuperHyperMatching. Thus it's a dual $1$-SuperHyperDefensive SuperHyperMatching.
\\
$(v),(vi)$ are obvious by $(iv).$
\end{proof}
\begin{proposition}
Let $ESHG:(V,E)$  be a neutrosophic SuperHyperGraph. Then the number is at most $\mathcal{O}(ESHG)$ and  the neutrosophic number is at most $\mathcal{O}_n(ESHG).$
\end{proposition}
\begin{proof}
Suppose $ESHG:(V,E)$ is a neutrosophic SuperHyperGraph. Consider $V.$ All SuperHyperMembers of $V$ have at least one SuperHyperNeighbor inside the SuperHyperSet more than SuperHyperNeighbor out of SuperHyperSet. Thus,
\\
$V$ is a dual SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in V,~|N(a)\cap V|>|N(a)\cap (V\setminus V)|   \equiv  \\&& \forall a\in V,~|N(a)\cap V|>|N(a)\cap \emptyset|   \equiv \\&&  \forall a\in V,~|N(a)\cap V|>|\emptyset|   \equiv  \\&& \forall a\in V,~|N(a)\cap V|>0   \equiv \\&&  \forall a\in V,~\delta>0.
\end{eqnarray*}
$V$ is a dual SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N_s(a)\cap S|>|N_s(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in V,~|N_s(a)\cap V|>|N_s(a)\cap (V\setminus V)|   \equiv  \\&& \forall a\in V,~|N_s(a)\cap V|>|N_s(a)\cap \emptyset|   \equiv \\&&  \forall a\in V,~|N_s(a)\cap V|>|\emptyset|   \equiv  \\&& \forall a\in V,~|N_s(a)\cap V|>0   \equiv \\&&  \forall a\in V,~\delta>0.
\end{eqnarray*}
$V$ is connected a dual SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N_c(a)\cap S|>|N_c(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in V,~|N_c(a)\cap V|>|N_c(a)\cap (V\setminus V)|   \equiv  \\&& \forall a\in V,~|N_c(a)\cap V|>|N_c(a)\cap \emptyset|   \equiv \\&&  \forall a\in V,~|N_c(a)\cap V|>|\emptyset|   \equiv  \\&& \forall a\in V,~|N_c(a)\cap V|>0   \equiv \\&&  \forall a\in V,~\delta>0.
\end{eqnarray*}
$V$ is a dual $\delta$-SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N(a)\cap S)-(N(a)\cap (V\setminus S))| > \delta  \equiv \\&&
 \forall a\in V,~|(N(a)\cap V)-(N(a)\cap (V\setminus V))| > \delta  \equiv \\&&
 \forall a\in V,~|(N(a)\cap V)-(N(a)\cap (\emptyset))| > \delta  \equiv \\&&
 \forall a\in V,~|(N(a)\cap V)-(\emptyset)| > \delta  \equiv \\&&
 \forall a\in V,~|(N(a)\cap V)| > \delta.
\end{eqnarray*}
$V$ is a dual strong $\delta$-SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N_s(a)\cap S)-(N_s(a)\cap (V\setminus S))| > \delta  \equiv \\&&
 \forall a\in V,~|(N_s(a)\cap V)-(N_s(a)\cap (V\setminus V))| > \delta  \equiv \\&&
 \forall a\in V,~|(N_s(a)\cap V)-(N_s(a)\cap (\emptyset))| > \delta  \equiv \\&&
 \forall a\in V,~|(N_s(a)\cap V)-(\emptyset)| > \delta  \equiv \\&&
 \forall a\in V,~|(N_s(a)\cap V)| > \delta.
\end{eqnarray*}
$V$ is a dual connected $\delta$-SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N_c(a)\cap S)-(N_c(a)\cap (V\setminus S))| > \delta  \equiv \\&&
 \forall a\in V,~|(N_c(a)\cap V)-(N_c(a)\cap (V\setminus V))| > \delta  \equiv \\&&
 \forall a\in V,~|(N_c(a)\cap V)-(N_c(a)\cap (\emptyset))| > \delta  \equiv \\&&
 \forall a\in V,~|(N_c(a)\cap V)-(\emptyset)| > \delta  \equiv \\&&
 \forall a\in V,~|(N_c(a)\cap V)| > \delta.
\end{eqnarray*}
Thus $V$ is a dual SuperHyperDefensive SuperHyperMatching and $V$ is the biggest SuperHyperSet in $ESHG:(V,E).$ Then the number is at most $\mathcal{O}(ESHG:(V,E))$ and  the neutrosophic number is at most $\mathcal{O}_n(ESHG:(V,E)).$
\end{proof}
\begin{proposition}
Let $ESHG:(V,E)$  be a neutrosophic SuperHyperGraph which is SuperHyperComplete. The number is $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and  the neutrosophic number is $\min\Sigma_{v\in\{v_1,v_2,\cdots,v_t\}_{t>\frac{\mathcal{O}(ESHG:(V,E))}{2}}\subseteq V}\sigma(v),$ in the setting of dual
\begin{itemize}
 \item[$(i):$]
 SuperHyperDefensive SuperHyperMatching;
  \item[$(ii):$] strong
  SuperHyperDefensive SuperHyperMatching;
    \item[$(iii):$] connected SuperHyperDefensive SuperHyperMatching;
      \item[$(iv):$]  $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperMatching;
        \item[$(v):$]  strong $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperMatching;
          \item[$(vi):$]  connected $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$
Consider $n$ half $-1$ SuperHyperVertices are out of $S$ which is a dual SuperHyperDefensive SuperHyperMatching. A SuperHyperVertex has $n$ half SuperHyperNeighbors in $S.$
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{n}{2}>\frac{n}{2}-1.  \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a dual SuperHyperDefensive SuperHyperMatching in a given SuperHyperComplete SuperHyperGraph. Thus the number is  $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and  the neutrosophic number is $\min\Sigma_{v\in\{v_1,v_2,\cdots,v_t\}_{t>\frac{\mathcal{O}(ESHG:(V,E))}{2}}\subseteq V}\sigma(v),$ in the setting of a dual SuperHyperDefensive SuperHyperMatching.
\\
$(ii).$
Consider $n$ half $-1$ SuperHyperVertices are out of $S$ which is a dual SuperHyperDefensive SuperHyperMatching. A SuperHyperVertex has $n$ half SuperHyperNeighbors in $S.$
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{n}{2}>\frac{n}{2}-1.  \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a dual strong SuperHyperDefensive SuperHyperMatching in a given SuperHyperComplete SuperHyperGraph. Thus the number is $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and  the neutrosophic number is $\min\Sigma_{v\in\{v_1,v_2,\cdots,v_t\}_{t>\frac{\mathcal{O}(ESHG:(V,E))}{2}}\subseteq V}\sigma(v),$ in the setting of a dual strong SuperHyperDefensive SuperHyperMatching.
\\
$(iii).$
Consider $n$ half $-1$ SuperHyperVertices are out of $S$ which is a dual SuperHyperDefensive SuperHyperMatching. A SuperHyperVertex has $n$ half SuperHyperNeighbors in $S.$
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{n}{2}>\frac{n}{2}-1.  
 \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a dual connected SuperHyperDefensive SuperHyperMatching in a given SuperHyperComplete SuperHyperGraph. Thus the number is  $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and  the neutrosophic number is $\min\Sigma_{v\in\{v_1,v_2,\cdots,v_t\}_{t>\frac{\mathcal{O}(ESHG:(V,E))}{2}}\subseteq V}\sigma(v),$ in the setting of a dual connected SuperHyperDefensive SuperHyperMatching.
\\
$(iv).$
Consider $n$ half $-1$ SuperHyperVertices are out of $S$ which is a dual SuperHyperDefensive SuperHyperMatching. A SuperHyperVertex has $n$ half SuperHyperNeighbors in $S.$
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{n}{2}>\frac{n}{2}-1.
 \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a dual  $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperMatching in a given SuperHyperComplete SuperHyperGraph. Thus the number is  $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and  the neutrosophic number is $\min\Sigma_{v\in\{v_1,v_2,\cdots,v_t\}_{t>\frac{\mathcal{O}(ESHG:(V,E))}{2}}\subseteq V}\sigma(v),$ in the setting of a dual  $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperMatching.
\\
$(v).$
Consider $n$ half $-1$ SuperHyperVertices are out of $S$ which is a dual SuperHyperDefensive SuperHyperMatching. A SuperHyperVertex has $n$ half SuperHyperNeighbors in $S.$
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{n}{2}>\frac{n}{2}-1.
 \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a dual strong $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperMatching in a given SuperHyperComplete SuperHyperGraph. Thus the number is  $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and  the neutrosophic number is $\min\Sigma_{v\in\{v_1,v_2,\cdots,v_t\}_{t>\frac{\mathcal{O}(ESHG:(V,E))}{2}}\subseteq V}\sigma(v),$ in the setting of a dual strong $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperMatching.
\\
$(vi).$
Consider $n$ half $-1$ SuperHyperVertices are out of $S$ which is a dual SuperHyperDefensive SuperHyperMatching. A SuperHyperVertex has $n$ half SuperHyperNeighbors in $S.$
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{n}{2}>\frac{n}{2}-1.  
 \end{eqnarray*}
 Thus it's proved. It implies every $S$ is a dual connected $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperMatching in a given SuperHyperComplete SuperHyperGraph. Thus the number is  $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and  the neutrosophic number is $\min\Sigma_{v\in\{v_1,v_2,\cdots,v_t\}_{t>\frac{\mathcal{O}(ESHG:(V,E))}{2}}\subseteq V}\sigma(v),$ in the setting of a dual connected $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperMatching.
\end{proof}
\begin{proposition}
Let $ESHG:(V,E)$ be a neutrosophic SuperHyperGraph which is $\emptyset.$ The number is  $0$ and  the neutrosophic number is $0,$ for an independent SuperHyperSet in the setting of dual
\begin{itemize}
 \item[$(i):$] SuperHyperDefensive SuperHyperMatching;
  \item[$(ii):$]  strong  SuperHyperDefensive SuperHyperMatching;
    \item[$(iii):$] connected  SuperHyperDefensive SuperHyperMatching;
      \item[$(iv):$]  $0$-SuperHyperDefensive SuperHyperMatching;
        \item[$(v):$]  strong $0$-SuperHyperDefensive SuperHyperMatching;
          \item[$(vi):$]  connected $0$-SuperHyperDefensive SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
Suppose $ESHG:(V,E)$ is a neutrosophic SuperHyperGraph. Consider $\emptyset.$ All SuperHyperMembers of $\emptyset$ have no SuperHyperNeighbor inside the SuperHyperSet less than SuperHyperNeighbor out of SuperHyperSet. Thus,
\\
$(i).$ $\emptyset$ is a dual  SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in \emptyset,~|N(a)\cap \emptyset|<|N(a)\cap (V\setminus \emptyset)|   \equiv  \\&& \forall a\in\emptyset,~|\emptyset|<|N(a)\cap (V\setminus \emptyset)|   \equiv \\&&  \forall a\in \emptyset,~0<|N(a)\cap V|   \equiv  \\&& \forall a\in \emptyset,~0<|N(a)\cap V|   \equiv \\&&  \forall a\in V,~\delta>0.
\end{eqnarray*}
The number is  $0$ and  the neutrosophic number is $0,$ for an independent SuperHyperSet in the setting of a dual SuperHyperDefensive SuperHyperMatching.
\\
$(ii).$ $\emptyset$ is a dual strong SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N_s(a)\cap S|<|N_s(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in \emptyset,~|N_s(a)\cap \emptyset|<|N_s(a)\cap (V\setminus \emptyset)|   \equiv  \\&& \forall a\in\emptyset,~|\emptyset|<|N_s(a)\cap (V\setminus \emptyset)|   \equiv \\&&  \forall a\in \emptyset,~0<|N_s(a)\cap V|   \equiv  \\&& \forall a\in \emptyset,~0<|N_s(a)\cap V|   \equiv \\&&  \forall a\in V,~\delta>0.
\end{eqnarray*}
The number is  $0$ and  the neutrosophic number is $0,$ for an independent SuperHyperSet in the setting of a dual strong  SuperHyperDefensive SuperHyperMatching.
\\
$(iii).$ $\emptyset$ is a dual connected SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|N_c(a)\cap S|<|N_c(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in \emptyset,~|N_c(a)\cap \emptyset|<|N_c(a)\cap (V\setminus \emptyset)|   \equiv  \\&& \forall a\in\emptyset,~|\emptyset|<|N_c(a)\cap (V\setminus \emptyset)|   \equiv \\&&  \forall a\in \emptyset,~0<|N_c(a)\cap V|   \equiv  \\&& \forall a\in \emptyset,~0<|N_c(a)\cap V|   \equiv \\&&  \forall a\in V,~\delta>0.
\end{eqnarray*}
The number is  $0$ and  the neutrosophic number is $0,$ for an independent SuperHyperSet in the setting of a dual connected SuperHyperDefensive SuperHyperMatching.
\\
$(iv).$ $\emptyset$ is a dual SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N(a)\cap S)-(N(a)\cap (V\setminus S))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|(N(a)\cap \emptyset)-(N(a)\cap (V\setminus \emptyset))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|(N(a)\cap \emptyset)-(N(a)\cap (V))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|\emptyset| < \delta  \equiv \\&&
 \forall a\in V,~0 < \delta.
\end{eqnarray*}
The number is  $0$ and  the neutrosophic number is $0,$ for an independent SuperHyperSet in the setting of a dual $0$-SuperHyperDefensive SuperHyperMatching.
\\
$(v).$ $\emptyset$ is a dual  strong $0$-SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N_s(a)\cap S)-(N_s(a)\cap (V\setminus S))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|(N_s(a)\cap \emptyset)-(N_s(a)\cap (V\setminus \emptyset))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|(N_s(a)\cap \emptyset)-(N_s(a)\cap (V))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|\emptyset| < \delta  \equiv \\&&
 \forall a\in V,~0 < \delta.
\end{eqnarray*}
The number is  $0$ and  the neutrosophic number is $0,$ for an independent SuperHyperSet in the setting of a dual strong $0$-SuperHyperDefensive SuperHyperMatching.
\\
$(vi).$ $\emptyset$ is a dual connected SuperHyperDefensive SuperHyperMatching since the following statements are equivalent.
 \begin{eqnarray*}&&
 \forall a\in S,~|(N_c(a)\cap S)-(N_c(a)\cap (V\setminus S))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|(N_c(a)\cap \emptyset)-(N_c(a)\cap (V\setminus \emptyset))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|(N_c(a)\cap \emptyset)-(N_c(a)\cap (V))| < \delta  \equiv \\&&
 \forall a\in \emptyset,~|\emptyset| < \delta  \equiv \\&&
 \forall a\in V,~0 < \delta.
\end{eqnarray*}
The number is  $0$ and  the neutrosophic number is $0,$ for an independent SuperHyperSet in the setting of a dual connected $0$-offensive SuperHyperDefensive SuperHyperMatching.
\end{proof}

\begin{proposition}
 Let $ESHG:(V,E)$ be a neutrosophic SuperHyperGraph which is SuperHyperComplete. Then there's no independent SuperHyperSet.
\end{proposition}

\begin{proposition}
Let $ESHG:(V,E)$ be a neutrosophic SuperHyperGraph which is SuperHyperCycle/SuperHyperPath/SuperHyperWheel. The number is  $\mathcal{O}(ESHG:(V,E))$ and  the neutrosophic number is $\mathcal{O}_n(ESHG:(V,E)),$ in the setting of a dual
\begin{itemize}
 \item[$(i):$]  SuperHyperDefensive SuperHyperMatching;
  \item[$(ii):$]  strong  SuperHyperDefensive SuperHyperMatching;
    \item[$(iii):$] connected  SuperHyperDefensive SuperHyperMatching;
      \item[$(iv):$]  $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyperMatching;
        \item[$(v):$]  strong $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyperMatching;
          \item[$(vi):$]  connected $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
Suppose $ESHG:(V,E)$  is a neutrosophic SuperHyperGraph which is SuperHyperCycle/SuperHyperPath/SuperHyperWheel. \\
$(i).$
Consider one SuperHyperVertex is out of $S$ which is a dual SuperHyperDefensive SuperHyperMatching. This SuperHyperVertex has one SuperHyperNeighbor in $S$, i.e, suppose $x\in V\setminus S$ such that $y,z\in N(x).$ By it's SuperHyperCycle, $|N(x)|=|N(y)|=|N(z)|=2.$ Thus
 \begin{eqnarray*}&&
 \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|   \equiv  \\&& \exists y\in V\setminus \{x\},~|N(y)\cap S|<|N(y)\cap (V\setminus (V\setminus \{x\}))|   \equiv \\&&  \exists y\in V\setminus \{x\},~|N(y)\cap S|<|N(y)\cap \{x\})|   \equiv \\&&  \exists y\in V\setminus \{x\},~|\{z\}|<|\{x\})|   \equiv \\&&  \exists y\in S,~1<1.  
\end{eqnarray*}
Thus it's contradiction. It implies every $V\setminus\{x\}$ isn't a dual  SuperHyperDefensive SuperHyperMatching in a given SuperHyperCycle.
\\
Consider one SuperHyperVertex is out of $S$ which is a dual  SuperHyperDefensive SuperHyperMatching. This SuperHyperVertex has one SuperHyperNeighbor in $S$, i.e, Suppose $x\in V\setminus S$ such that $y,z\in N(x).$ By it's SuperHyperPath, $|N(x)|=|N(y)|=|N(z)|=2.$ Thus
 \begin{eqnarray*}&&
 \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|   \equiv  \\&& \exists y\in V\setminus \{x\},~|N(y)\cap S|<|N(y)\cap (V\setminus (V\setminus \{x\}))|   \equiv \\&&  \exists y\in V\setminus \{x\},~|N(y)\cap S|<|N(y)\cap \{x\})|   \equiv \\&&  \exists y\in V\setminus \{x\},~|\{z\}|<|\{x\})|   \equiv \\&&  \exists y\in S,~1<1.  
\end{eqnarray*}
Thus it's contradiction. It implies every $V\setminus\{x\}$ isn't a dual SuperHyperDefensive SuperHyperMatching in a given SuperHyperPath.
\\
Consider one SuperHyperVertex is out of $S$ which is a dual SuperHyperDefensive SuperHyperMatching. This SuperHyperVertex has one SuperHyperNeighbor in $S$, i.e, Suppose $x\in V\setminus S$ such that $y,z\in N(x).$ By it's SuperHyperWheel, $|N(x)|=|N(y)|=|N(z)|=2.$ Thus
 \begin{eqnarray*}&&
 \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|    \equiv \\&&  \forall a\in S,~|N(a)\cap S|<|N(a)\cap (V\setminus S)|   \equiv  \\&& \exists y\in V\setminus \{x\},~|N(y)\cap S|<|N(y)\cap (V\setminus (V\setminus \{x\}))|   \equiv \\&&  \exists y\in V\setminus \{x\},~|N(y)\cap S|<|N(y)\cap \{x\})|   \equiv \\&&  \exists y\in V\setminus \{x\},~|\{z\}|<|\{x\})|   \equiv \\&&  \exists y\in S,~1<1.  
\end{eqnarray*}
Thus it's contradiction. It implies every $V\setminus\{x\}$ isn't a dual  SuperHyperDefensive SuperHyperMatching in a given SuperHyperWheel.
\\
$(ii),$ $(iii)$ are obvious by $(i).$
\\
$(iv).$ By $(i),$ $V$ is maximal and it's  a dual  SuperHyperDefensive SuperHyperMatching. Thus it's  a dual $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyperMatching.
\\
$(v),(vi)$ are obvious by $(iv).$
\\
Thus the number is  $\mathcal{O}(ESHG:(V,E))$ and  the neutrosophic number is $\mathcal{O}_n(ESHG:(V,E)),$ in the setting of all types of a dual SuperHyperDefensive SuperHyperMatching.
\end{proof}
\begin{proposition}
Let $ESHG:(V,E)$  be a neutrosophic SuperHyperGraph which is SuperHyperStar/complete SuperHyperBipartite/complete SuperHyperMultiPartite. The number is  $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and  the neutrosophic number is $\min\Sigma_{v\in\{v_1,v_2,\cdots,v_t\}_{t>\frac{\mathcal{O}(ESHG:(V,E))}{2}}\subseteq V}\sigma(v),$ in the setting of a dual
\begin{itemize}
 \item[$(i):$]  SuperHyperDefensive SuperHyperMatching;
  \item[$(ii):$]  strong  SuperHyperDefensive SuperHyperMatching;
    \item[$(iii):$] connected  SuperHyperDefensive SuperHyperMatching;
      \item[$(iv):$]  $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperMatching;
        \item[$(v):$]  strong $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperMatching;
          \item[$(vi):$]  connected $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$
 Consider $n$ half $+1$ SuperHyperVertices are in $S$ which is SuperHyperDefensive SuperHyperMatching. A SuperHyperVertex has at most $n$ half SuperHyperNeighbors in $S.$ If the SuperHyperVertex is the non-SuperHyperCenter, then
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~1>0.  
\end{eqnarray*}
If the SuperHyperVertex is the SuperHyperCenter, then
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{n}{2}>\frac{n}{2}-1.  
 \end{eqnarray*}
Thus it's proved. It implies every $S$ is a dual  SuperHyperDefensive SuperHyperMatching in a given SuperHyperStar.
 \\
 Consider $n$ half $+1$ SuperHyperVertices are in $S$ which is a dual SuperHyperDefensive SuperHyperMatching.
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{\delta}{2}>n-\frac{\delta}{2}.  
 \end{eqnarray*}
Thus it's proved. It implies every $S$ is  a dual  SuperHyperDefensive SuperHyperMatching in a given complete SuperHyperBipartite which isn't a SuperHyperStar.
 \\
 Consider $n$ half $+1$ SuperHyperVertices are in $S$ which is a dual SuperHyperDefensive SuperHyperMatching and they are chosen from different SuperHyperParts, equally or almost equally as possible. A SuperHyperVertex in $S$ has $\delta$ half SuperHyperNeighbors in $S.$
 \begin{eqnarray*}&&
  \forall a\in S,~|N(a)\cap S|>|N(a)\cap (V\setminus S)|
  \equiv \\&&  \forall a\in S,~\frac{\delta}{2}>n-\frac{\delta}{2}.  
 \end{eqnarray*}
\\
Thus it's proved. It implies every $S$ is a dual  SuperHyperDefensive SuperHyperMatching in a given complete SuperHyperMultipartite which is neither a SuperHyperStar nor complete SuperHyperBipartite.
\\
$(ii),$ $(iii)$ are obvious by $(i).$
\\
$(iv).$ By $(i),$ $\{x_i\}_{i=1}^{\frac{\mathcal{O}(ESHG:(V,E))}{2}+1}$ is maximal and it's a dual  SuperHyperDefensive SuperHyperMatching. Thus it's a dual $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$-SuperHyperDefensive SuperHyperMatching.
\\
$(v),(vi)$ are obvious by $(iv).$
\\
Thus the number is  $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and  the neutrosophic number is $\min\Sigma_{v\in\{v_1,v_2,\cdots,v_t\}_{t>\frac{\mathcal{O}(ESHG:(V,E))}{2}}\subseteq V}\sigma(v),$ in the setting of all dual SuperHyperMatching.
\end{proof}
\begin{proposition}
 Let $\mathcal{NSHF}:(V,E)$ be a SuperHyperFamily of the $ESHGs:(V,E)$ neutrosophic SuperHyperGraphs which are from one-type SuperHyperClass which the result is obtained for the individuals. Then the results also hold for the SuperHyperFamily $\mathcal{NSHF}:(V,E)$ of these specific SuperHyperClasses of the  neutrosophic SuperHyperGraphs.
\end{proposition}
\begin{proof}
 There are neither SuperHyperConditions nor SuperHyperRestrictions on the SuperHyperVertices. Thus the SuperHyperResults on individuals, $ESHGs:(V,E),$ are extended to the SuperHyperResults on SuperHyperFamily, $\mathcal{NSHF}:(V,E).$
\end{proof}
%
\begin{proposition}
  Let $ESHG:(V,E)$  be a strong neutrosophic SuperHyperGraph. If $S$ is a dual SuperHyperDefensive SuperHyperMatching, then $\forall v\in V\setminus S,~\exists x\in S$ such that
   \begin{itemize}
\item[$(i)$]
$v\in N_s(x);$
\item[$(ii)$]
$vx\in E.$
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$ is a strong neutrosophic SuperHyperGraph. Consider $v\in V\setminus S.$ Since $S$ is a dual SuperHyperDefensive SuperHyperMatching,
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)| \\&& v\in V\setminus S,~|N_s(v)\cap S|>|N_s(v)\cap (V\setminus S)|
\\&& v\in V\setminus S, \exists x\in S,~v\in N_s(x).
\end{eqnarray*}
 $(ii).$ Suppose $ESHG:(V,E)$ is a strong neutrosophic SuperHyperGraph. Consider $v\in V\setminus S.$ Since $S$ is a dual SuperHyperDefensive SuperHyperMatching,
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)| \\&& v\in V\setminus S,~|N_s(v)\cap S|>|N_s(v)\cap (V\setminus S)|
\\&& v\in V\setminus S, \exists x\in S:~v\in N_s(x)
\\&& v\in V\setminus S, \exists x\in S: vx\in E,~\mu(vx)=\sigma(v)\wedge\sigma(x).
\\&& v\in V\setminus S, \exists x\in S: vx\in E.
\end{eqnarray*}
\end{proof}
\begin{proposition}
   Let $ESHG:(V,E)$  be a strong neutrosophic SuperHyperGraph. If $S$ is a dual SuperHyperDefensive SuperHyperMatching, then
    \begin{itemize}
\item[$(i)$]
$S$ is SuperHyperDominating set;
\item[$(ii)$]
there's $S\subseteq S'$ such that $|S'|$ is SuperHyperChromatic number.
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$ is a strong neutrosophic SuperHyperGraph. Consider $v\in V\setminus S.$ Since $S$ is a dual  SuperHyperDefensive SuperHyperMatching, either
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)| \\&& v\in V\setminus S,~|N_s(v)\cap S|>|N_s(v)\cap (V\setminus S)|
\\&& v\in V\setminus S, \exists x\in S,~v\in N_s(x)
\end{eqnarray*}
or
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)| \\&& v\in V\setminus S,~|N_s(v)\cap S|>|N_s(v)\cap (V\setminus S)|
\\&& v\in V\setminus S, \exists x\in S:~v\in N_s(x)
\\&& v\in V\setminus S, \exists x\in S: vx\in E,~\mu(vx)=\sigma(v)\wedge\sigma(x)
\\&& v\in V\setminus S, \exists x\in S: vx\in E.
\end{eqnarray*}
It implies $S$ is SuperHyperDominating SuperHyperSet.
\\
 $(ii).$ Suppose $ESHG:(V,E)$ is a strong neutrosophic SuperHyperGraph. Consider $v\in V\setminus S.$ Since $S$ is a dual  SuperHyperDefensive SuperHyperMatching, either
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)| \\&& v\in V\setminus S,~|N_s(v)\cap S|>|N_s(v)\cap (V\setminus S)|
\\&& v\in V\setminus S, \exists x\in S,~v\in N_s(x)
\end{eqnarray*}
or
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)| \\&& v\in V\setminus S,~|N_s(v)\cap S|>|N_s(v)\cap (V\setminus S)|
\\&& v\in V\setminus S, \exists x\in S:~v\in N_s(x)
\\&& v\in V\setminus S, \exists x\in S: vx\in E,~\mu(vx)=\sigma(v)\wedge\sigma(x)
\\&& v\in V\setminus S, \exists x\in S: vx\in E.
\end{eqnarray*}
Thus every SuperHyperVertex $v\in V\setminus S,$ has at least one SuperHyperNeighbor in $S.$ The only case is about the relation amid SuperHyperVertices in $S$ in the terms of SuperHyperNeighbors.
 It implies there's $S\subseteq S'$ such that $|S'|$ is SuperHyperChromatic number.
 \end{proof}
\begin{proposition}
  Let $ESHG:(V,E)$  be a strong neutrosophic SuperHyperGraph. Then
      \begin{itemize}
\item[$(i)$]
$\Gamma\leq\mathcal{O};$
\item[$(ii)$]
$\Gamma_s\leq\mathcal{O}_n.$
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$ is a strong neutrosophic SuperHyperGraph. Let $S=V.$
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)| \\&& v\in V\setminus V,~|N_s(v)\cap V|>|N_s(v)\cap (V\setminus V)|
\\&& v\in \emptyset,~|N_s(v)\cap V|>|N_s(v)\cap \emptyset|
\\&& v\in \emptyset,~|N_s(v)\cap V|>|\emptyset|
\\&& v\in \emptyset,~|N_s(v)\cap V|>0
\end{eqnarray*}
It implies $V$ is a dual SuperHyperDefensive SuperHyperMatching. For all SuperHyperSets of SuperHyperVertices $S,~S\subseteq V.$ Thus for all SuperHyperSets of SuperHyperVertices $S,~|S|\leq |V|.$ It implies for all SuperHyperSets of SuperHyperVertices $S,~|S|\leq \mathcal{O}.$ So for all SuperHyperSets of SuperHyperVertices $S,~\Gamma\leq \mathcal{O}.$
\\
$(ii).$ Suppose $ESHG:(V,E)$ is a strong neutrosophic SuperHyperGraph. Let $S=V.$
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)| \\&& v\in V\setminus V,~|N_s(v)\cap V|>|N_s(v)\cap (V\setminus V)|
\\&& v\in \emptyset,~|N_s(v)\cap V|>|N_s(v)\cap \emptyset|
\\&& v\in \emptyset,~|N_s(v)\cap V|>|\emptyset|
\\&& v\in \emptyset,~|N_s(v)\cap V|>0
\end{eqnarray*}
It implies $V$ is a dual SuperHyperDefensive SuperHyperMatching. For all SuperHyperSets of neutrosophic SuperHyperVertices $S,~S\subseteq V.$ Thus for all SuperHyperSets of neutrosophic SuperHyperVertices $S,~\Sigma_{s\in S}\Sigma_{i=1}^{3}\sigma_i(s)\leq \Sigma_{v\in V}\Sigma_{i=1}^{3}\sigma_i(v).$ It implies for all SuperHyperSets of neutrosophic SuperHyperVertices $S,~\Sigma_{s\in S}\Sigma_{i=1}^{3}\sigma_i(s)\leq \mathcal{O}_n.$ So for all SuperHyperSets of neutrosophic SuperHyperVertices $S,~\Gamma_s\leq \mathcal{O}_n.$
\end{proof}
\begin{proposition}
  Let $ESHG:(V,E)$  be a strong neutrosophic SuperHyperGraph which is connected. Then
      \begin{itemize}
\item[$(i)$]
$\Gamma\leq\mathcal{O}-1;$
\item[$(ii)$]
$\Gamma_s\leq\mathcal{O}_n-\Sigma_{i=1}^{3}\sigma_i(x).$
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$ is a strong neutrosophic SuperHyperGraph. Let $S=V-\{x\}$ where $x$ is arbitrary and $x\in V.$
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)| \\&& v\in V\setminus V-\{x\},~|N_s(v)\cap (V-\{x\})|>|N_s(v)\cap (V\setminus (V-\{x\}))|
\\&& ~|N_s(x)\cap (V-\{x\})|>|N_s(x)\cap \{x\}|
\\&& ~|N_s(x)\cap (V-\{x\})|>|\emptyset|
\\&& ~|N_s(x)\cap (V-\{x\})|>0
\end{eqnarray*}
It implies $V-\{x\}$ is a dual  SuperHyperDefensive SuperHyperMatching. For all SuperHyperSets of SuperHyperVertices $S\neq V,~S\subseteq V-\{x\}.$ Thus for all SuperHyperSets of SuperHyperVertices $S\neq V,~|S|\leq |V-\{x\}|.$ It implies for all SuperHyperSets of SuperHyperVertices $S\neq V,~|S|\leq \mathcal{O}-1.$ So for all SuperHyperSets of SuperHyperVertices $S,~\Gamma\leq \mathcal{O}-1.$
\\
$(ii).$ Suppose $ESHG:(V,E)$ is a strong neutrosophic SuperHyperGraph. Let $S=V-\{x\}$ where $x$ is arbitrary and $x\in V.$
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)| \\&& v\in V\setminus V-\{x\},~|N_s(v)\cap (V-\{x\})|>|N_s(v)\cap (V\setminus (V-\{x\}))|
\\&& ~|N_s(x)\cap (V-\{x\})|>|N_s(x)\cap \{x\}|
\\&& ~|N_s(x)\cap (V-\{x\})|>|\emptyset|
\\&& ~|N_s(x)\cap (V-\{x\})|>0
\end{eqnarray*}
It implies $V-\{x\}$ is a dual SuperHyperDefensive SuperHyperMatching. For all SuperHyperSets of neutrosophic SuperHyperVertices $S\neq V,~S\subseteq V-\{x\}.$ Thus for all SuperHyperSets of neutrosophic SuperHyperVertices $S\neq V,~\Sigma_{s\in S}\Sigma_{i=1}^{3}\sigma_i(s)\leq \Sigma_{v\in V-\{x\}}\Sigma_{i=1}^{3}\sigma_i(v).$ It implies for all SuperHyperSets of neutrosophic SuperHyperVertices $S\neq V,~\Sigma_{s\in S}\Sigma_{i=1}^{3}\sigma_i(s)\leq \mathcal{O}_n-\Sigma_{i=1}^{3}\sigma_i(x).$ So for all SuperHyperSets of neutrosophic SuperHyperVertices $S,~\Gamma_s\leq \mathcal{O}_n-\Sigma_{i=1}^{3}\sigma_i(x).$
\end{proof}

\begin{proposition}
  Let $ESHG:(V,E)$  be an odd SuperHyperPath. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n-1}\}$ is  a dual SuperHyperDefensive SuperHyperMatching;
\item[$(ii)$]
 $\Gamma=\lfloor\frac{n}{2}\rfloor+1$ and corresponded SuperHyperSet is $S=\{v_2,v_4,\cdots,v_{n-1}\}$;
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S=\{v_2,v_4,\cdots,v_{n-1}\}}\Sigma_{i=1}^3\sigma_i(s), \Sigma_{s\in S=\{v_1,v_3,\cdots,v_{n-1}\}}\Sigma_{i=1}^3\sigma_i(s)\};$
\item[$(iv)$]
the SuperHyperSets $S_1=\{v_2,v_4,\cdots,v_{n-1}\}$ and $S_2=\{v_1,v_3,\cdots,v_{n-1}\}$ are only a dual SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$ is an odd SuperHyperPath. Let $S=\{v_2,v_4,\cdots,v_{n-1}\}$ where for all $v_i,v_j\in\{v_2,v_4,\cdots,v_{n-1}\},~v_iv_j\not\in E$ and $v_i,v_j\in V.$
 \begin{eqnarray*}&&
  v\in \{v_1,v_3,\cdots,v_{n}\},~|N_s(v)\cap \{v_2,v_4,\cdots.v_{n-1}\}|=2>\\&&0=|N_s(v)\cap \{v_1,v_3,\cdots,v_{n}\}|
 \forall z\in V\setminus S,~|N_s(z)\cap S|=2>\\&&0=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|
\\&& v\in V\setminus \{v_2,v_4,\cdots,v_{n-1}\},~|N_s(v)\cap \{v_2,v_4,\cdots.v_{n-1}\}|>\\&&|N_s(v)\cap (V\setminus\{v_2,v_4,\cdots.v_{n-1}\})|
\end{eqnarray*}
It implies $S=\{v_2,v_4,\cdots,v_{n-1}\}$  is a dual SuperHyperDefensive SuperHyperMatching. If $S=\{v_2,v_4,\cdots,v_{n-1}\}-\{v_i\}$ where $v_i\in\{v_2,v_4,\cdots,v_{n-1}\},$ then
 \begin{eqnarray*}&&   
 \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1=1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1\not>1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|\not>|N_s(z)\cap (V\setminus S)|.
\end{eqnarray*}
So $\{v_2,v_4,\cdots,v_{n-1}\}-\{v_i\}$ where $v_i\in\{v_2,v_4,\cdots,v_{n-1}\}$ isn't a dual SuperHyperDefensive SuperHyperMatching. It induces $S=\{v_2,v_4,\cdots,v_{n-1}\}$ is a dual SuperHyperDefensive SuperHyperMatching.
\\
$(ii)$ and $(iii)$ are trivial.
\\
$(iv).$ By $(i),$ $S_1=\{v_2,v_4,\cdots,v_{n-1}\}$ is a dual SuperHyperDefensive SuperHyperMatching. Thus it's enough to show that $S_2=\{v_1,v_3,\cdots,v_{n-1}\}$ is a dual SuperHyperDefensive SuperHyperMatching.
Suppose $ESHG:(V,E)$ is an odd SuperHyperPath. Let $S=\{v_1,v_3,\cdots,v_{n-1}\}$ where for all $v_i,v_j\in\{v_1,v_3,\cdots,v_{n-1}\},~v_iv_j\not\in E$ and $v_i,v_j\in V.$
 \begin{eqnarray*}&&
  v\in \{v_2,v_4,\cdots,v_{n}\},~|N_s(v)\cap \{v_1,v_3,\cdots.v_{n-1}\}|=2>\\&&0=|N_s(v)\cap \{v_2,v_4,\cdots,v_{n}\}|
 \forall z\in V\setminus S,~|N_s(z)\cap S|=2>0=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|
\\&& v\in V\setminus \{v_1,v_3,\cdots,v_{n-1}\},~|N_s(v)\cap \{v_1,v_3,\cdots.v_{n-1}\}|>\\&&|N_s(v)\cap (V\setminus\{v_1,v_3,\cdots.v_{n-1}\})|
\end{eqnarray*}
It implies $S=\{v_1,v_3,\cdots,v_{n-1}\}$  is a dual SuperHyperDefensive SuperHyperMatching. If $S=\{v_1,v_3,\cdots,v_{n-1}\}-\{v_i\}$ where $v_i\in\{v_1,v_3,\cdots,v_{n-1}\},$ then
 \begin{eqnarray*}&&   
 \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1=1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1\not>1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|\not>|N_s(z)\cap (V\setminus S)|.
\end{eqnarray*}
So $\{v_1,v_3,\cdots,v_{n-1}\}-\{v_i\}$ where $v_i\in\{v_1,v_3,\cdots,v_{n-1}\}$ isn't a dual SuperHyperDefensive SuperHyperMatching. It induces $S=\{v_1,v_3,\cdots,v_{n-1}\}$ is a dual SuperHyperDefensive SuperHyperMatching.
\end{proof}
\begin{proposition}
  Let $ESHG:(V,E)$  be an even SuperHyperPath. Then
 \begin{itemize}
\item[$(i)$]
the set $S=\{v_2,v_4,\cdots.v_{n}\}$ is  a dual SuperHyperDefensive SuperHyperMatching;
\item[$(ii)$]
 $\Gamma=\lfloor\frac{n}{2}\rfloor$ and corresponded SuperHyperSets are $\{v_2,v_4,\cdots.v_n\}$ and $\{v_1,v_3,\cdots.v_{n-1}\};$
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S=\{v_2,v_4,\cdots,v_n\}}\Sigma_{i=1}^3\sigma_i(s), \Sigma_{s\in S=\{v_1,v_3,\cdots.v_{n-1}\}}\Sigma_{i=1}^3\sigma_i(s)\};$
\item[$(iv)$]
the SuperHyperSets $S_1=\{v_2,v_4,\cdots.v_{n}\}$ and $S_2=\{v_1,v_3,\cdots.v_{n-1}\}$ are only dual SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$ is an even SuperHyperPath. Let $S=\{v_2,v_4,\cdots,v_{n}\}$ where for all $v_i,v_j\in\{v_2,v_4,\cdots,v_{n}\},~v_iv_j\not\in E$ and $v_i,v_j\in V.$
 \begin{eqnarray*}&&
  v\in \{v_1,v_3,\cdots,v_{n-1}\},~|N_s(v)\cap \{v_2,v_4,\cdots.v_{n}\}|=2>\\&&0=|N_s(v)\cap \{v_1,v_3,\cdots,v_{n-1}\}|
 \forall z\in V\setminus S,~|N_s(z)\cap S|=2>\\&&0=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|
\\&& v\in V\setminus \{v_2,v_4,\cdots,v_{n}\},~|N_s(v)\cap \{v_2,v_4,\cdots.v_{n}\}|>|N_s(v)\cap (V\setminus\{v_2,v_4,\cdots.v_{n}\})|
\end{eqnarray*}
It implies $S=\{v_2,v_4,\cdots,v_{n}\}$  is a dual SuperHyperDefensive SuperHyperMatching. If $S=\{v_2,v_4,\cdots,v_{n}\}-\{v_i\}$ where $v_i\in\{v_2,v_4,\cdots,v_{n}\},$ then
 \begin{eqnarray*}&&   
 \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1=1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1\not>1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|\not>|N_s(z)\cap (V\setminus S)|.
\end{eqnarray*}
So $\{v_2,v_4,\cdots,v_{n}\}-\{v_i\}$ where $v_i\in\{v_2,v_4,\cdots,v_{n}\}$ isn't a dual SuperHyperDefensive SuperHyperMatching. It induces $S=\{v_2,v_4,\cdots,v_{n}\}$ is a dual SuperHyperDefensive SuperHyperMatching.
\\
$(ii)$ and $(iii)$ are trivial.
\\
$(iv).$ By $(i),$ $S_1=\{v_2,v_4,\cdots,v_{n}\}$ is a dual SuperHyperDefensive SuperHyperMatching. Thus it's enough to show that $S_2=\{v_1,v_3,\cdots,v_{n-1}\}$ is a dual SuperHyperDefensive SuperHyperMatching.
Suppose $ESHG:(V,E)$ is an even SuperHyperPath. Let $S=\{v_1,v_3,\cdots,v_{n-1}\}$ where for all $v_i,v_j\in\{v_1,v_3,\cdots,v_{n-1}\},~v_iv_j\not\in E$ and $v_i,v_j\in V.$
 \begin{eqnarray*}&&
  v\in \{v_2,v_4,\cdots,v_{n}\},~|N_s(v)\cap \{v_1,v_3,\cdots.v_{n-1}\}|=2>\\&&0=|N_s(v)\cap \{v_2,v_4,\cdots,v_{n}\}|
 \forall z\in V\setminus S,~|N_s(z)\cap S|=2>0=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|
\\&& v\in V\setminus \{v_1,v_3,\cdots,v_{n-1}\},~|N_s(v)\cap \{v_1,v_3,\cdots.v_{n-1}\}|>\\&&|N_s(v)\cap (V\setminus\{v_1,v_3,\cdots.v_{n-1}\})|
\end{eqnarray*}
It implies $S=\{v_1,v_3,\cdots,v_{n-1}\}$  is a dual SuperHyperDefensive SuperHyperMatching. If $S=\{v_1,v_3,\cdots,v_{n-1}\}-\{v_i\}$ where $v_i\in\{v_1,v_3,\cdots,v_{n-1}\},$ then
 \begin{eqnarray*}&&   
 \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1=1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1\not>1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|\not>|N_s(z)\cap (V\setminus S)|.
\end{eqnarray*}
So $\{v_1,v_3,\cdots,v_{n-1}\}-\{v_i\}$ where $v_i\in\{v_1,v_3,\cdots,v_{n-1}\}$ isn't a dual SuperHyperDefensive SuperHyperMatching. It induces $S=\{v_1,v_3,\cdots,v_{n-1}\}$ is a dual SuperHyperDefensive SuperHyperMatching.
\end{proof}
\begin{proposition}
  Let $ESHG:(V,E)$  be an even SuperHyperCycle. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n}\}$ is  a dual SuperHyperDefensive SuperHyperMatching;
\item[$(ii)$]
 $\Gamma=\lfloor\frac{n}{2}\rfloor$ and corresponded SuperHyperSets are $\{v_2,v_4,\cdots,v_n\}$ and $\{v_1,v_3,\cdots,v_{n-1}\};$
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S=\{v_2,v_4,\cdots,v_n\}}\sigma(s), \Sigma_{s\in S=\{v_1,v_3,\cdots,v_{n-1}\}}\sigma(s)\};$
\item[$(iv)$]
the SuperHyperSets $S_1=\{v_2,v_4,\cdots,v_{n}\}$ and $S_2=\{v_1,v_3,\cdots,v_{n-1}\}$ are only dual  SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$ is an even SuperHyperCycle. Let $S=\{v_2,v_4,\cdots,v_{n}\}$ where for all $v_i,v_j\in\{v_2,v_4,\cdots,v_{n}\},~v_iv_j\not\in E$ and $v_i,v_j\in V.$
 \begin{eqnarray*}&&
  v\in \{v_1,v_3,\cdots,v_{n-1}\},~|N_s(v)\cap \{v_2,v_4,\cdots.v_{n}\}|=2>\\&&0=|N_s(v)\cap \{v_1,v_3,\cdots,v_{n-1}\}|
 \forall z\in V\setminus S,~|N_s(z)\cap S|=2>\\&&0=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|
\\&& v\in V\setminus \{v_2,v_4,\cdots,v_{n}\},~|N_s(v)\cap \{v_2,v_4,\cdots.v_{n}\}|>\\&&|N_s(v)\cap (V\setminus\{v_2,v_4,\cdots.v_{n}\})|
\end{eqnarray*}
It implies $S=\{v_2,v_4,\cdots,v_{n}\}$  is  a dual  SuperHyperDefensive SuperHyperMatching. If $S=\{v_2,v_4,\cdots,v_{n}\}-\{v_i\}$ where $v_i\in\{v_2,v_4,\cdots,v_{n}\},$ then
 \begin{eqnarray*}&&   
 \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1=1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1\not>1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|\not>|N_s(z)\cap (V\setminus S)|.
\end{eqnarray*}
So $\{v_2,v_4,\cdots,v_{n}\}-\{v_i\}$ where $v_i\in\{v_2,v_4,\cdots,v_{n}\}$ isn't a  dual SuperHyperDefensive SuperHyperMatching. It induces $S=\{v_2,v_4,\cdots,v_{n}\}$ is  a dual  SuperHyperDefensive SuperHyperMatching.
\\
$(ii)$ and $(iii)$ are trivial.
\\
$(iv).$ By $(i),$ $S_1=\{v_2,v_4,\cdots,v_{n}\}$ is a  dual SuperHyperDefensive SuperHyperMatching. Thus it's enough to show that $S_2=\{v_1,v_3,\cdots,v_{n-1}\}$ is a  dual  SuperHyperDefensive SuperHyperMatching.
Suppose $ESHG:(V,E)$ is an even SuperHyperCycle. Let $S=\{v_1,v_3,\cdots,v_{n-1}\}$ where for all $v_i,v_j\in\{v_1,v_3,\cdots,v_{n-1}\},~v_iv_j\not\in E$ and $v_i,v_j\in V.$
 \begin{eqnarray*}&&
  v\in \{v_2,v_4,\cdots,v_{n}\},~|N_s(v)\cap \{v_1,v_3,\cdots.v_{n-1}\}|=2>\\&&0=|N_s(v)\cap \{v_2,v_4,\cdots,v_{n}\}|
 \forall z\in V\setminus S,~|N_s(z)\cap S|=2>0=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|
\\&& v\in V\setminus \{v_1,v_3,\cdots,v_{n-1}\},~|N_s(v)\cap \{v_1,v_3,\cdots.v_{n-1}\}|>\\&&|N_s(v)\cap (V\setminus\{v_1,v_3,\cdots.v_{n-1}\})|
\end{eqnarray*}
It implies $S=\{v_1,v_3,\cdots,v_{n-1}\}$  is a  dual SuperHyperDefensive SuperHyperMatching. If $S=\{v_1,v_3,\cdots,v_{n-1}\}-\{v_i\}$ where $v_i\in\{v_1,v_3,\cdots,v_{n-1}\},$ then
 \begin{eqnarray*}&&   
 \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1=1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1\not>1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|\not>|N_s(z)\cap (V\setminus S)|.
\end{eqnarray*}
So $\{v_1,v_3,\cdots,v_{n-1}\}-\{v_i\}$ where $v_i\in\{v_1,v_3,\cdots,v_{n-1}\}$ isn't a dual SuperHyperDefensive SuperHyperMatching. It induces $S=\{v_1,v_3,\cdots,v_{n-1}\}$ is a  dual  SuperHyperDefensive SuperHyperMatching.
\end{proof}
\begin{proposition}
  Let $ESHG:(V,E)$  be an odd SuperHyperCycle. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n-1}\}$ is  a  dual  SuperHyperDefensive SuperHyperMatching;
\item[$(ii)$]
 $\Gamma=\lfloor\frac{n}{2}\rfloor+1$ and corresponded SuperHyperSet is $S=\{v_2,v_4,\cdots,v_{n-1}\}$;
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S=\{v_2,v_4,\cdots.v_{n-1}\}}\Sigma_{i=1}^3\sigma_i(s), \Sigma_{s\in S=\{v_1,v_3,\cdots.v_{n-1}\}}\Sigma_{i=1}^3\sigma_i(s)\};$
\item[$(iv)$]
the SuperHyperSets $S_1=\{v_2,v_4,\cdots.v_{n-1}\}$ and $S_2=\{v_1,v_3,\cdots.v_{n-1}\}$ are only dual SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$ is an odd SuperHyperCycle. Let $S=\{v_2,v_4,\cdots,v_{n-1}\}$ where for all $v_i,v_j\in\{v_2,v_4,\cdots,v_{n-1}\},~v_iv_j\not\in E$ and $v_i,v_j\in V.$
 \begin{eqnarray*}&&
  v\in \{v_1,v_3,\cdots,v_{n}\},~|N_s(v)\cap \{v_2,v_4,\cdots.v_{n-1}\}|=2>\\&&0=|N_s(v)\cap \{v_1,v_3,\cdots,v_{n}\}|
 \forall z\in V\setminus S,~|N_s(z)\cap S|=2>0=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|
\\&& v\in V\setminus \{v_2,v_4,\cdots,v_{n-1}\},~|N_s(v)\cap \{v_2,v_4,\cdots.v_{n-1}\}|>\\&&|N_s(v)\cap (V\setminus\{v_2,v_4,\cdots.v_{n-1}\})|
\end{eqnarray*}
It implies $S=\{v_2,v_4,\cdots,v_{n-1}\}$  is  a dual SuperHyperDefensive SuperHyperMatching. If $S=\{v_2,v_4,\cdots,v_{n-1}\}-\{v_i\}$ where $v_i\in\{v_2,v_4,\cdots,v_{n-1}\},$ then
 \begin{eqnarray*}&&   
 \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1=1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1\not>1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|\not>|N_s(z)\cap (V\setminus S)|.
\end{eqnarray*}
So $\{v_2,v_4,\cdots,v_{n-1}\}-\{v_i\}$ where $v_i\in\{v_2,v_4,\cdots,v_{n-1}\}$ isn't a dual SuperHyperDefensive SuperHyperMatching. It induces $S=\{v_2,v_4,\cdots,v_{n-1}\}$ is a dual SuperHyperDefensive SuperHyperMatching.
\\
$(ii)$ and $(iii)$ are trivial.
\\
$(iv).$ By $(i),$ $S_1=\{v_2,v_4,\cdots,v_{n-1}\}$ is a dual SuperHyperDefensive SuperHyperMatching. Thus it's enough to show that $S_2=\{v_1,v_3,\cdots,v_{n-1}\}$ is a dual SuperHyperDefensive SuperHyperMatching.
Suppose $ESHG:(V,E)$ is an odd SuperHyperCycle. Let $S=\{v_1,v_3,\cdots,v_{n-1}\}$ where for all $v_i,v_j\in\{v_1,v_3,\cdots,v_{n-1}\},~v_iv_j\not\in E$ and $v_i,v_j\in V.$
 \begin{eqnarray*}&&
  v\in \{v_2,v_4,\cdots,v_{n}\},~|N_s(v)\cap \{v_1,v_3,\cdots.v_{n-1}\}|=2>\\&&0=|N_s(v)\cap \{v_2,v_4,\cdots,v_{n}\}|
 \forall z\in V\setminus S,~|N_s(z)\cap S|=2>0=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|
\\&& v\in V\setminus \{v_1,v_3,\cdots,v_{n-1}\},~|N_s(v)\cap \{v_1,v_3,\cdots.v_{n-1}\}|>\\&&|N_s(v)\cap (V\setminus\{v_1,v_3,\cdots.v_{n-1}\})|
\end{eqnarray*}
It implies $S=\{v_1,v_3,\cdots,v_{n-1}\}$  is  a dual SuperHyperDefensive SuperHyperMatching. If $S=\{v_1,v_3,\cdots,v_{n-1}\}-\{v_i\}$ where $v_i\in\{v_1,v_3,\cdots,v_{n-1}\},$ then
 \begin{eqnarray*}&&   
 \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1=1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|=1\not>1=|N_s(z)\cap (V\setminus S)|
\\&& \exists v_{i+1}\in V\setminus S,~|N_s(z)\cap S|\not>|N_s(z)\cap (V\setminus S)|.
\end{eqnarray*}
So $\{v_1,v_3,\cdots,v_{n-1}\}-\{v_i\}$ where $v_i\in\{v_1,v_3,\cdots,v_{n-1}\}$ isn't a dual SuperHyperDefensive SuperHyperMatching. It induces $S=\{v_1,v_3,\cdots,v_{n-1}\}$ is a dual SuperHyperDefensive SuperHyperMatching.
\end{proof}
\begin{proposition}
  Let $ESHG:(V,E)$  be SuperHyperStar. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{c\}$ is  a dual maximal SuperHyperMatching;
\item[$(ii)$]
$\Gamma=1;$
\item[$(iii)$]
$\Gamma_s=\Sigma_{i=1}^3\sigma_i(c);$
\item[$(iv)$]
the SuperHyperSets $S=\{c\}$ and $S\subset S'$ are only dual SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$  is a SuperHyperStar.
 \begin{eqnarray*}&&
  \forall v\in V\setminus\{c\},~|N_s(v)\cap \{c\}|=1>\\&&0=|N_s(v)\cap (V\setminus\{c\})|
 \forall z\in V\setminus S,~|N_s(z)\cap S|=1>\\&&0=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|
\\&& v\in V\setminus\{c\},~|N_s(v)\cap \{c\}|>|N_s(v)\cap (V\setminus\{c\})|
\end{eqnarray*}
It implies $S=\{c\}$  is a dual SuperHyperDefensive SuperHyperMatching. If $S=\{c\}-\{c\}=\emptyset,$ then
 \begin{eqnarray*}&&   
 \exists v\in V\setminus S,~|N_s(z)\cap S|=0=0=|N_s(z)\cap (V\setminus S)|
\\&& \exists v\in V\setminus S,~|N_s(z)\cap S|=0\not>0=|N_s(z)\cap (V\setminus S)|
\\&& \exists v\in V\setminus S,~|N_s(z)\cap S|\not>|N_s(z)\cap (V\setminus S)|.
\end{eqnarray*}
So $S=\{c\}-\{c\}=\emptyset$  isn't a dual  SuperHyperDefensive SuperHyperMatching. It induces $S=\{c\}$ is a dual SuperHyperDefensive SuperHyperMatching.
\\
$(ii)$ and $(iii)$ are trivial.
\\
$(iv).$ By $(i),$ $S=\{c\}$ is a dual SuperHyperDefensive SuperHyperMatching. Thus it's enough to show that $S\subseteq S'$ is a dual  SuperHyperDefensive SuperHyperMatching.
Suppose $ESHG:(V,E)$ is a SuperHyperStar. Let $S\subseteq S'.$
 \begin{eqnarray*}&&
  \forall v\in V\setminus\{c\},~|N_s(v)\cap \{c\}|=1>\\&&0=|N_s(v)\cap (V\setminus\{c\})|
 \forall z\in V\setminus S',~|N_s(z)\cap S'|=1>\\&&0=|N_s(z)\cap (V\setminus S')|
\\&& \forall z\in V\setminus S',~|N_s(z)\cap S'|>|N_s(z)\cap (V\setminus S')|
\end{eqnarray*}
It implies $S'\subseteq S$  is a dual SuperHyperDefensive SuperHyperMatching.
\end{proof}
\begin{proposition}
  Let $ESHG:(V,E)$  be SuperHyperWheel. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n}$ is a dual maximal  SuperHyperDefensive SuperHyperMatching;
\item[$(ii)$]
$\Gamma=|\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n}|;$
\item[$(iii)$]
$\Gamma_s=\Sigma_{\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n}}\Sigma_{i=1}^3\sigma_i(s);$
\item[$(iv)$]
the SuperHyperSet $\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n}$ is only a dual maximal  SuperHyperDefensive SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$ is a SuperHyperWheel. Let $S=\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n}.$ There are either
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|=2>1=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|
\end{eqnarray*}
or
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|=3>0=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|
\end{eqnarray*}
It implies $S=\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n}$ is a dual  SuperHyperDefensive SuperHyperMatching. If $S'=\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n}-\{z\}$ where $z\in S=\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n},$ then
There are either
 \begin{eqnarray*}&&
 \forall z\in V\setminus S',~|N_s(z)\cap S'|=1<2=|N_s(z)\cap (V\setminus S')|
\\&& \forall z\in V\setminus S',~|N_s(z)\cap S'|<|N_s(z)\cap (V\setminus S')|
\\&& \forall z\in V\setminus S',~|N_s(z)\cap S'|\not>|N_s(z)\cap (V\setminus S')|
\end{eqnarray*}
or
 \begin{eqnarray*}&&
 \forall z\in V\setminus S',~|N_s(z)\cap S'|=1=1=|N_s(z)\cap (V\setminus S')|
\\&& \forall z\in V\setminus S',~|N_s(z)\cap S'|=|N_s(z)\cap (V\setminus S')|
\\&& \forall z\in V\setminus S',~|N_s(z)\cap S'|\not>|N_s(z)\cap (V\setminus S')|
\end{eqnarray*}
So $S'=\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n}-\{z\}$ where $z\in S=\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n}$ isn't a dual  SuperHyperDefensive SuperHyperMatching. It induces $S=\{v_1,v_3\}\cup\{v_6,v_9\cdots,v_{i+6},\cdots,v_n\}_{i=1}^{6+3(i-1)\leq n}$ is a dual maximal SuperHyperDefensive SuperHyperMatching.
\\
$(ii),$ $(iii)$ and $(iv)$ are obvious.
\end{proof}
\begin{proposition}
  Let $ESHG:(V,E)$  be an odd SuperHyperComplete. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$ is  a dual SuperHyperDefensive SuperHyperMatching;
\item[$(ii)$]
$\Gamma=\lfloor\frac{n}{2}\rfloor+1;$
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S}\Sigma_{i=1}^3\sigma_i(s)\}_{S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}};$
\item[$(iv)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$ is only a dual  SuperHyperDefensive SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$ is an odd SuperHyperComplete. Let $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}.$ Thus  
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|=\lfloor\frac{n}{2}\rfloor+1>\lfloor\frac{n}{2}\rfloor-1=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|
\end{eqnarray*}
It implies $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$  is a dual SuperHyperDefensive SuperHyperMatching. If $S'=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1} -\{z\}$ where $z\in S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1},$ then
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|=\lfloor\frac{n}{2}\rfloor=\lfloor\frac{n}{2}\rfloor=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|\not>|N_s(z)\cap (V\setminus S)|
\end{eqnarray*}
So $S'=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1} -\{z\}$ where $z\in S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$ isn't a dual SuperHyperDefensive SuperHyperMatching. It induces $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$ is a dual SuperHyperDefensive SuperHyperMatching.
\\
$(ii),$ $(iii)$ and $(iv)$ are obvious.
\end{proof}

\begin{proposition}
  Let $ESHG:(V,E)$  be an even SuperHyperComplete. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ is a dual SuperHyperDefensive SuperHyperMatching;
\item[$(ii)$]
$\Gamma=\lfloor\frac{n}{2}\rfloor;$
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S}\Sigma_{i=1}^3\sigma_i(s)\}_{S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}};$
\item[$(iv)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ is only a dual maximal SuperHyperDefensive SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$ is an even SuperHyperComplete. Let $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}.$ Thus  
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|=\lfloor\frac{n}{2}\rfloor>\lfloor\frac{n}{2}\rfloor-1=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|.
\end{eqnarray*}
It implies $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$  is a dual SuperHyperDefensive SuperHyperMatching. If $S'=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor} -\{z\}$ where $z\in S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor},$ then
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|=\lfloor\frac{n}{2}\rfloor-1<\lfloor\frac{n}{2}\rfloor+1=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|\not>|N_s(z)\cap (V\setminus S)|.
\end{eqnarray*}
So $S'=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor} -\{z\}$ where $z\in S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ isn't a dual SuperHyperDefensive SuperHyperMatching. It induces $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ is a dual maximal SuperHyperDefensive SuperHyperMatching.
\\
$(ii),$ $(iii)$ and $(iv)$ are obvious.
\end{proof}

\begin{proposition}
  Let $\mathcal{NSHF}:(V,E)$ be a $m$-SuperHyperFamily of neutrosophic SuperHyperStars with common neutrosophic SuperHyperVertex SuperHyperSet. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{c_1,c_2,\cdots,c_m\}$ is  a dual SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF};$
\item[$(ii)$]
$\Gamma=m$ for $\mathcal{NSHF}:(V,E);$
\item[$(iii)$]
$\Gamma_s=\Sigma_{i=1}^m\Sigma_{j=1}^3\sigma_j(c_i)$ for $\mathcal{NSHF}:(V,E);$
\item[$(iv)$]
the SuperHyperSets $S=\{c_1,c_2,\cdots,c_m\}$ and $S\subset S'$ are only dual  SuperHyperMatching for $\mathcal{NSHF}:(V,E).$
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$ is a SuperHyperStar.
 \begin{eqnarray*}&&
  \forall v\in V\setminus\{c\},~|N_s(v)\cap \{c\}|=1>\\&&0=|N_s(v)\cap (V\setminus\{c\})|
 \forall z\in V\setminus S,~|N_s(z)\cap S|=1>\\&&0=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|
\\&& v\in V\setminus\{c\},~|N_s(v)\cap \{c\}|>|N_s(v)\cap (V\setminus\{c\})|
\end{eqnarray*}
It implies $S=\{c_1,c_2,\cdots,c_m\}$  is a dual SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF}:(V,E).$ If $S=\{c\}-\{c\}=\emptyset,$ then
 \begin{eqnarray*}&&   
 \exists v\in V\setminus S,~|N_s(z)\cap S|=0=0=|N_s(z)\cap (V\setminus S)|
\\&& \exists v\in V\setminus S,~|N_s(z)\cap S|=0\not>0=|N_s(z)\cap (V\setminus S)|
\\&& \exists v\in V\setminus S,~|N_s(z)\cap S|\not>|N_s(z)\cap (V\setminus S)|.
\end{eqnarray*}
So $S=\{c\}-\{c\}=\emptyset$  isn't a dual  SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF}:(V,E).$ It induces $S=\{c_1,c_2,\cdots,c_m\}$ is a dual maximal  SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF}:(V,E).$
\\
$(ii)$ and $(iii)$ are trivial.
\\
$(iv).$ By $(i),$ $S=\{c_1,c_2,\cdots,c_m\}$ is a dual SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF}:(V,E).$  Thus it's enough to show that $S\subseteq S'$ is a dual  SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF}:(V,E).$
Suppose $ESHG:(V,E)$  is a SuperHyperStar. Let $S\subseteq S'.$
 \begin{eqnarray*}&&
  \forall v\in V\setminus\{c\},~|N_s(v)\cap \{c\}|=1>\\&&0=|N_s(v)\cap (V\setminus\{c\})|
 \forall z\in V\setminus S',~|N_s(z)\cap S'|=1>\\&&0=|N_s(z)\cap (V\setminus S')|
\\&& \forall z\in V\setminus S',~|N_s(z)\cap S'|>|N_s(z)\cap (V\setminus S')|
\end{eqnarray*}
It implies $S'\subseteq S$  is a dual SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF}:(V,E).$
\end{proof}

\begin{proposition}
  Let $\mathcal{NSHF}:(V,E)$ be an $m$-SuperHyperFamily of odd SuperHyperComplete SuperHyperGraphs with common neutrosophic SuperHyperVertex SuperHyperSet. Then
\begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$ is  a dual maximal SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF};$
\item[$(ii)$]
$\Gamma=\lfloor\frac{n}{2}\rfloor+1$ for $\mathcal{NSHF}:(V,E);$
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S}\Sigma_{i=1}^3\sigma_i(s)\}_{S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}}$ for $\mathcal{NSHF}:(V,E);$
\item[$(iv)$]
the SuperHyperSets $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$ are only a dual maximal SuperHyperMatching for $\mathcal{NSHF}:(V,E).$
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$ is odd SuperHyperComplete. Let $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}.$ Thus  
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|=\lfloor\frac{n}{2}\rfloor+1>\lfloor\frac{n}{2}\rfloor-1=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|
\end{eqnarray*}
It implies $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$  is a dual SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF}:(V,E).$ If $S'=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1} -\{z\}$ where $z\in S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1},$ then
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|=\lfloor\frac{n}{2}\rfloor=\lfloor\frac{n}{2}\rfloor=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|\not>|N_s(z)\cap (V\setminus S)|
\end{eqnarray*}
So $S'=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1} -\{z\}$ where $z\in S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$ isn't a dual SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF}:(V,E).$ It induces $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$ is a dual maximal  SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF}:(V,E).$
\\
$(ii),$ $(iii)$ and $(iv)$ are obvious.
\end{proof}

\begin{proposition}
  Let $\mathcal{NSHF}:(V,E)$ be a $m$-SuperHyperFamily of even SuperHyperComplete SuperHyperGraphs with common neutrosophic SuperHyperVertex SuperHyperSet. Then
\begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ is  a dual SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF}:(V,E);$
\item[$(ii)$]
$\Gamma=\lfloor\frac{n}{2}\rfloor$ for $\mathcal{NSHF}:(V,E);$
\item[$(iii)$]
$\Gamma_s=\min\{\Sigma_{s\in S}\Sigma_{i=1}^3\sigma_i(s)\}_{S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}}$ for $\mathcal{NSHF}:(V,E);$
\item[$(iv)$]
the SuperHyperSets $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ are only dual  maximal SuperHyperMatching for $\mathcal{NSHF}:(V,E).$
\end{itemize}
\end{proposition}
\begin{proof}
 $(i).$ Suppose $ESHG:(V,E)$ is even SuperHyperComplete. Let $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}.$ Thus  
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|=\lfloor\frac{n}{2}\rfloor>\lfloor\frac{n}{2}\rfloor-1=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|>|N_s(z)\cap (V\setminus S)|.
\end{eqnarray*}
It implies $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$  is a dual SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF}:(V,E).$ If $S'=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor} -\{z\}$ where $z\in S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor},$ then
 \begin{eqnarray*}&&
 \forall z\in V\setminus S,~|N_s(z)\cap S|=\lfloor\frac{n}{2}\rfloor-1<\lfloor\frac{n}{2}\rfloor+1=|N_s(z)\cap (V\setminus S)|
\\&& \forall z\in V\setminus S,~|N_s(z)\cap S|\not>|N_s(z)\cap (V\setminus S)|.
\end{eqnarray*}
So $S'=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor} -\{z\}$ where $z\in S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ isn't a dual  SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF}:(V,E).$ It induces $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ is a dual maximal SuperHyperDefensive SuperHyperMatching for $\mathcal{NSHF}:(V,E).$
\\
$(ii),$ $(iii)$ and $(iv)$ are obvious.
\end{proof}

%

\begin{proposition}
   Let $ESHG:(V,E)$ be a strong neutrosophic SuperHyperGraph. Then following statements hold;
\begin{itemize}
 \item[$(i)$] if $s\geq t$ and a SuperHyperSet $S$ of SuperHyperVertices is an t-SuperHyperDefensive SuperHyperMatching, then $S$ is an s-SuperHyperDefensive SuperHyperMatching;
  \item[$(ii)$] if $s\leq t$ and a SuperHyperSet $S$ of SuperHyperVertices is a dual t-SuperHyperDefensive SuperHyperMatching, then $S$ is a dual s-SuperHyperDefensive  SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
$(i).$ Suppose $ESHG:(V,E)$ is a strong neutrosophic SuperHyperGraph. Consider a SuperHyperSet $S$ of SuperHyperVertices is an t-SuperHyperDefensive  SuperHyperMatching. Then
 \begin{eqnarray*}&&
 \forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<t; \\&& \forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<t\leq s;
\\&& \forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<s.
\end{eqnarray*}
Thus $S$ is an s-SuperHyperDefensive  SuperHyperMatching.
\\
$(ii).$
Suppose $ESHG:(V,E)$ is a strong neutrosophic SuperHyperGraph. Consider a SuperHyperSet $S$ of SuperHyperVertices is a dual t-SuperHyperDefensive  SuperHyperMatching. Then
 \begin{eqnarray*}&&
 \forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>t; \\&& \forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>t\geq s;
\\&& \forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>s.
\end{eqnarray*}
Thus $S$ is a dual s-SuperHyperDefensive  SuperHyperMatching.
\end{proof}

\begin{proposition}
   Let $ESHG:(V,E)$  be a strong neutrosophic SuperHyperGraph. Then following statements hold;
\begin{itemize}
 \item[$(i)$] if $s\geq t+2$ and a SuperHyperSet $S$ of SuperHyperVertices is an t-SuperHyperDefensive  SuperHyperMatching, then $S$ is an s-SuperHyperPowerful  SuperHyperMatching;
  \item[$(ii)$] if $s\leq t$ and a SuperHyperSet $S$ of SuperHyperVertices is a dual t-SuperHyperDefensive SuperHyperMatching, then $S$ is a dual s-SuperHyperPowerful  SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
$(i).$ Suppose $ESHG:(V,E)$ is a strong neutrosophic SuperHyperGraph. Consider a SuperHyperSet $S$ of SuperHyperVertices is an t-SuperHyperDefensive  SuperHyperMatching. Then
\begin{eqnarray*}
&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<t;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<t\leq t+2\leq s;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<s.
\end{eqnarray*}
Thus $S$ is an $(t+2)-$SuperHyperDefensive SuperHyperMatching. By $S$ is an $s-$SuperHyperDefensive SuperHyperMatching and $S$ is a dual $(s+2)-$SuperHyperDefensive  SuperHyperMatching, $S$ is an s-SuperHyperPowerful   SuperHyperMatching.
\\
$(ii).$
Suppose $ESHG:(V,E)$ is a strong neutrosophic SuperHyperGraph. Consider a SuperHyperSet $S$ of SuperHyperVertices is a dual t-SuperHyperDefensive  SuperHyperMatching. Then
\begin{eqnarray*}
&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>t;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>t\geq s>s-2;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>s-2.
\end{eqnarray*}
Thus $S$ is an $(s-2)-$SuperHyperDefensive  SuperHyperMatching. By $S$ is an $(s-2)-$SuperHyperDefensive SuperHyperMatching and $S$ is a dual $s-$SuperHyperDefensive SuperHyperMatching, $S$ is an $s-$SuperHyperPowerful SuperHyperMatching.
\end{proof}

\begin{proposition}
   Let $ESHG:(V,E)$  be a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph. Then following statements hold;
\begin{itemize}
 \item[$(i)$] if $\forall a\in S,~|N_s(a)\cap S|< \lfloor \frac{r}{2}\rfloor+1,$ then $ESHG:(V,E)$ is an 2-SuperHyperDefensive SuperHyperMatching;
  \item[$(ii)$]  if $\forall a\in V\setminus S,~|N_s(a)\cap S|> \lfloor \frac{r}{2}\rfloor+1,$ then $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive SuperHyperMatching;
   \item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is an r-SuperHyperDefensive  SuperHyperMatching;
     \item[$(iv)$]  if $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is a dual r-SuperHyperDefensive SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
$(i).$ Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph. Then
\begin{eqnarray*}
&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<\lfloor \frac{r}{2}\rfloor+1-(\lfloor \frac{r}{2}\rfloor-1);
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<\lfloor \frac{r}{2}\rfloor+1-(\lfloor \frac{r}{2}\rfloor-1)<2;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<2.
\end{eqnarray*}
Thus $S$ is an 2-SuperHyperDefensive  SuperHyperMatching.
\\
$(ii).$
Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph. Then
\begin{eqnarray*}
&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>\lfloor \frac{r}{2}\rfloor+1-(\lfloor \frac{r}{2}\rfloor-1);
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>\lfloor \frac{r}{2}\rfloor+1-(\lfloor \frac{r}{2}\rfloor-1)>2;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>2.
\end{eqnarray*}
Thus $S$ is a dual 2-SuperHyperDefensive  SuperHyperMatching.
\\
$(iii).$ Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph. Then
\begin{eqnarray*}
&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<r-0;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<r-0=r;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<r.
\end{eqnarray*}
Thus $S$ is an r-SuperHyperDefensive SuperHyperMatching.
\\
$(iv).$
Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph. Then
\begin{eqnarray*}
&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>r-0;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>r-0=r;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>r.
\end{eqnarray*}
Thus $S$ is a dual r-SuperHyperDefensive SuperHyperMatching.
\end{proof}
\begin{proposition}
   Let $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph. Then following statements hold;
\begin{itemize}
 \item[$(i)$] $\forall a\in S,~|N_s(a)\cap S|< \lfloor \frac{r}{2}\rfloor+1$ if $ESHG:(V,E)$ is an 2-SuperHyperDefensive  SuperHyperMatching;
  \item[$(ii)$]  $\forall a\in V\setminus S,~|N_s(a)\cap S|> \lfloor \frac{r}{2}\rfloor+1$ if $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyperMatching;
   \item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an r-SuperHyperDefensive SuperHyperMatching;
     \item[$(iv)$]  $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is a dual r-SuperHyperDefensive  SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
$(i).$ Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph. Then
\begin{eqnarray*}
&& \forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<2;\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<2=\lfloor \frac{r}{2}\rfloor+1-(\lfloor \frac{r}{2}\rfloor-1);
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<\lfloor \frac{r}{2}\rfloor+1-(\lfloor \frac{r}{2}\rfloor-1);\\&&
\forall t\in S,~|N_s(t)\cap S|=\lfloor \frac{r}{2}\rfloor+1,~ |N_s(t)\cap(V\setminus S)|=\lfloor \frac{r}{2}\rfloor-1.
\end{eqnarray*}
$(ii).$
Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph and a dual 2-SuperHyperDefensive  SuperHyperMatching. Then
\begin{eqnarray*}
&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|> 2;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>2=\lfloor \frac{r}{2}\rfloor+1-(\lfloor \frac{r}{2}\rfloor-1);
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>\lfloor \frac{r}{2}\rfloor+1-(\lfloor \frac{r}{2}\rfloor-1);
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|=\lfloor \frac{r}{2}\rfloor+1,~|N_s(t)\cap (V\setminus S)=\lfloor \frac{r}{2}\rfloor-1.
\end{eqnarray*}
$(iii).$  Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph and an r-SuperHyperDefensive  SuperHyperMatching.
\begin{eqnarray*}
&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<r;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<r=r-0;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<r-0;
\\&&\forall t\in S,~|N_s(t)\cap S|=r,~|N_s(t)\cap (V\setminus S)|=0.
\end{eqnarray*}
$(iv).$
Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph and a dual r-SuperHyperDefensive  SuperHyperMatching. Then
\begin{eqnarray*}
&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>r;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>r=r-0;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>r-0;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|=r,~|N_s(t)\cap (V\setminus S)|=0.
\end{eqnarray*}
\end{proof}

\begin{proposition}
   Let $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph which is a  SuperHyperComplete. Then following statements hold;
\begin{itemize}
 \item[$(i)$] $\forall a\in S,~|N_s(a)\cap S|< \lfloor \frac{\mathcal{O}-1}{2}\rfloor+1$ if $ESHG:(V,E)$ is an 2-SuperHyperDefensive  SuperHyperMatching;
  \item[$(ii)$]  $\forall a\in V\setminus S,~|N_s(a)\cap S|> \lfloor \frac{\mathcal{O}-1}{2}\rfloor+1$ if $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyperMatching;
   \item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an $(\mathcal{O}-1)$-SuperHyperDefensive  SuperHyperMatching;
     \item[$(iv)$]  $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is a dual $(\mathcal{O}-1)$-SuperHyperDefensive  SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
$(i).$ Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph and an 2- SuperHyperDefensive  SuperHyperMatching. Then
\begin{eqnarray*}
&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<2;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<2=\lfloor \frac{\mathcal{O}-1}{2}\rfloor+1-(\lfloor \frac{\mathcal{O}-1}{2}\rfloor-1);
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<\lfloor \frac{\mathcal{O}-1}{2}\rfloor+1-(\lfloor \frac{\mathcal{O}-1}{2}\rfloor-1);
\\&&\forall t\in S,~|N_s(t)\cap S|=\lfloor \frac{\mathcal{O}-1}{2}\rfloor+1,~ |N_s(t)\cap(V\setminus S)|=\lfloor \frac{\mathcal{O}-1}{2}\rfloor-1.
\end{eqnarray*}
$(ii).$
Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph and a dual 2-SuperHyperDefensive  SuperHyperMatching. Then
\begin{eqnarray*}
&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|> 2;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>2=\lfloor \frac{\mathcal{O}-1}{2}\rfloor+1-(\lfloor \frac{\mathcal{O}-1}{2}\rfloor-1);
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>\lfloor \frac{\mathcal{O}-1}{2}\rfloor+1-(\lfloor \frac{\mathcal{O}-1}{2}\rfloor-1);
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|=\lfloor \frac{\mathcal{O}-1}{2}\rfloor+1,~|N_s(t)\cap (V\setminus S)=\lfloor \frac{\mathcal{O}-1}{2}\rfloor-1.
\end{eqnarray*}
$(iii).$  Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph and an ($\mathcal{O}-1$)-SuperHyperDefensive  SuperHyperMatching.
\begin{eqnarray*}
&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<\mathcal{O}-1;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<\mathcal{O}-1=\mathcal{O}-1-0;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<\mathcal{O}-1-0;
\\&&\forall t\in S,~|N_s(t)\cap S|=\mathcal{O}-1,~|N_s(t)\cap (V\setminus S)|=0.
\end{eqnarray*}
$(iv).$
Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph and a dual r-SuperHyperDefensive  SuperHyperMatching. Then
\begin{eqnarray*}
&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>\mathcal{O}-1;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>\mathcal{O}-1=\mathcal{O}-1-0;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>\mathcal{O}-1-0;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|=\mathcal{O}-1,~|N_s(t)\cap (V\setminus S)|=0.
\end{eqnarray*}
\end{proof}

\begin{proposition}
   Let $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph which is a SuperHyperComplete. Then following statements hold;
\begin{itemize}
 \item[$(i)$] if $\forall a\in S,~|N_s(a)\cap S|< \lfloor \frac{\mathcal{O}-1}{2}\rfloor+1,$ then $ESHG:(V,E)$ is an 2-SuperHyperDefensive  SuperHyperMatching;
  \item[$(ii)$]  if $\forall a\in V\setminus S,~|N_s(a)\cap S|> \lfloor \frac{\mathcal{O}-1}{2}\rfloor+1,$ then $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyperMatching;
   \item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is $(\mathcal{O}-1)$-SuperHyperDefensive  SuperHyperMatching;
     \item[$(iv)$]  if $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is a dual $(\mathcal{O}-1)$-SuperHyperDefensive  SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
$(i).$ Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph which is a SuperHyperComplete. Then
\begin{eqnarray*}
&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<\lfloor \frac{\mathcal{O}-1}{2}\rfloor+1-(\lfloor \frac{\mathcal{O}-1}{2}\rfloor-1);
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<\lfloor \frac{\mathcal{O}-1}{2}\rfloor+1-(\lfloor \frac{\mathcal{O}-1}{2}\rfloor-1)<2;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<2.
\end{eqnarray*}
Thus $S$ is an 2-SuperHyperDefensive  SuperHyperMatching.
\\
$(ii).$
Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph which is a SuperHyperComplete. Then
\begin{eqnarray*}
&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>\lfloor \frac{\mathcal{O}-1}{2}\rfloor+1-(\lfloor \frac{\mathcal{O}-1}{2}\rfloor-1);
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>\lfloor \frac{\mathcal{O}-1}{2}\rfloor+1-(\lfloor \frac{\mathcal{O}-1}{2}\rfloor-1)>2;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>2.
\end{eqnarray*}
Thus $S$ is a dual 2-SuperHyperDefensive  SuperHyperMatching.
\\
$(iii).$ Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph which is a SuperHyperComplete. Then
\begin{eqnarray*}
&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<\mathcal{O}-1-0;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<\mathcal{O}-1-0=\mathcal{O}-1;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<\mathcal{O}-1.
\end{eqnarray*}
Thus $S$ is an ($\mathcal{O}-1$)-SuperHyperDefensive  SuperHyperMatching.
\\
$(iv).$
Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph which is a SuperHyperComplete. Then
\begin{eqnarray*}
&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>\mathcal{O}-1-0;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>\mathcal{O}-1-0=\mathcal{O}-1;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>\mathcal{O}-1.
\end{eqnarray*}
Thus $S$ is a dual ($\mathcal{O}-1$)-SuperHyperDefensive  SuperHyperMatching.
\end{proof}

\begin{proposition}
   Let $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph which is SuperHyperCycle. Then following statements hold;
\begin{itemize}
 \item[$(i)$] $\forall a\in S,~|N_s(a)\cap S|< 2$ if $ESHG:(V,E))$ is an 2-SuperHyperDefensive  SuperHyperMatching;
  \item[$(ii)$]  $\forall a\in V\setminus S,~|N_s(a)\cap S|> 2$ if $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyperMatching;
   \item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an 2-SuperHyperDefensive  SuperHyperMatching;
     \item[$(iv)$]  $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyperMatching.
\end{itemize}
\end{proposition}
\begin{proof}
$(i).$ Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph and $S$ is an 2-SuperHyperDefensive  SuperHyperMatching. Then
\begin{eqnarray*}
&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<2;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<2=2-0;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<2;
\\&&\forall t\in S,~|N_s(t)\cap S|<2,~ |N_s(t)\cap(V\setminus S)|=0.
\end{eqnarray*}
$(ii).$
Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph and $S$ is  a dual 2-SuperHyperDefensive  SuperHyperMatching. Then
\begin{eqnarray*}
&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|> 2;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>2=2-0;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>2;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|>2,~|N_s(t)\cap (V\setminus S)=0.
\end{eqnarray*}
$(iii).$  Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph and $S$ is an 2-SuperHyperDefensive  SuperHyperMatching.
\begin{eqnarray*}
&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<2;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<2=2-0;
\\&&\forall t\in S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|<2-0;
\\&&\forall t\in S,~|N_s(t)\cap S|<2,~|N_s(t)\cap (V\setminus S)|=0.
\end{eqnarray*}
$(iv).$
Suppose $ESHG:(V,E)$ is a[an] [r-]SuperHyperUniform-strong-neutrosophic SuperHyperGraph and $S$ is a dual r-SuperHyperDefensive  SuperHyperMatching. Then
\begin{eqnarray*}
&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>2;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>2=2-0;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|-|N_s(t)\cap (V\setminus S)|>2-0;
\\&&\forall t\in V\setminus S,~|N_s(t)\cap S|>2,~|N_s(t)\cap (V\setminus S)|=0.
\end{eqnarray*}
\end{proof}