New Ideas On Super Nodes By Hyper Moves Of Tree-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph

“#177 Article”

Henry Garrett, “New Ideas On Super Nodes By Hyper Moves Of Tree-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.32825.24163).

@ResearchGate: https://www.researchgate.net/publication/369030046
@Scribd: https://www.scribd.com/document/-
@ZENODO_ORG: https://zenodo.org/record/-
@academia: https://www.academia.edu/98024333

 

\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}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhf{}
\fancyhead[LE,RO]{Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA }
\fancyfoot[LE,RO]{Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA }
% document begins here
\begin{document}
\vspace*{0.35in}
\linenumbers
% title goes here:
\begin{flushleft}
{\Large
\textbf\newline{
New Ideas On Super Nodes By Hyper Moves Of Tree-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph
}
}
\newline
\newline
Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA
% authors go here:

\end{flushleft}
\section{ABSTRACT}
In this scientific research, (Different Neutrosophic Types of Neutrosophic SuperHyperTree-Decomposition). Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is an Tree-Decomposition pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called Neutrosophic e-SuperHyperTree-Decomposition if it's an ordered pair $(T,\mathcal{T})$ where $T$ is a tree and $\mathcal{T}$ is a family of subtrees of $T$ such that  $T_a\cap T_b\neq \empty$ if $ab\in E;$     Neutrosophic re-SuperHyperTree-Decomposition if it's an ordered pair $(T,\mathcal{T})$ where $T$ is a tree and $\mathcal{T}$ is a family of subtrees of $T$ such that  $T_a\cap T_b\neq \empty$ if $ab\in E;$  and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$ Neutrosophic v-SuperHyperTree-Decomposition if it's an ordered pair $(T,\mathcal{T})$ where $T$ is a tree and $\mathcal{T}$ is a family of subtrees of $T$ such that  $T_a\cap T_b\neq \empty$ if $ab\in E;$  Neutrosophic rv-SuperHyperTree-Decomposition if it's an ordered pair $(T,\mathcal{T})$ where $T$ is a tree and $\mathcal{T}$ is a family of subtrees of $T$ such that  $T_a\cap T_b\neq \empty$ if $ab\in E;$  and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$ Neutrosophic SuperHyperTree-Decomposition if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition. ((Neutrosophic) SuperHyperTree-Decomposition).
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called an Extreme SuperHyperTree-Decomposition if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTree-Decomposition; a Neutrosophic SuperHyperTree-Decomposition if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ for a  Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality  consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperTree-Decomposition; an Extreme SuperHyperTree-Decomposition SuperHyperPolynomial  if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTree-Decomposition; and the Extreme power is corresponded to its Extreme coefficient; a Neutrosophic SuperHyperTree-Decomposition SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ 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 the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperTree-Decomposition; and the Neutrosophic power is corresponded to its Neutrosophic coefficient; an Extreme V-SuperHyperTree-Decomposition if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTree-Decomposition; a Neutrosophic V-SuperHyperTree-Decomposition if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperTree-Decomposition; an Extreme V-SuperHyperTree-Decomposition SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTree-Decomposition; and the Extreme power is corresponded to its Extreme coefficient; a Neutrosophic SuperHyperTree-Decomposition SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ 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 the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperTree-Decomposition; and the Neutrosophic power is corresponded to its Neutrosophic coefficient.  In this scientific research, new setting is introduced for new SuperHyperNotions, namely, a SuperHyperTree-Decomposition  and Neutrosophic SuperHyperTree-Decomposition. 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. Assume a SuperHyperGraph. Then $\delta-$SuperHyperTree-Decomposition 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 $\delta-$SuperHyperTree-Decomposition 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 SuperHyperTree-Decomposition . Since there's more ways to get type-results to make a SuperHyperTree-Decomposition  more understandable. For the sake of having Neutrosophic SuperHyperTree-Decomposition, there's a need to ``redefine'' the notion of a ``SuperHyperTree-Decomposition ''. 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 SuperHyperTree-Decomposition . It's redefined a Neutrosophic SuperHyperTree-Decomposition  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 SuperHyperTree-Decomposition . 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 SuperHyperTree-Decomposition until the SuperHyperTree-Decomposition, then it's officially called a ``SuperHyperTree-Decomposition'' but otherwise, it isn't a SuperHyperTree-Decomposition . There are some instances about the clarifications for the main definition titled a ``SuperHyperTree-Decomposition ''. These two examples get more scrutiny and discernment since there are characterized in the disciplinary ways of the SuperHyperClass based on a SuperHyperTree-Decomposition . For the sake of having a Neutrosophic SuperHyperTree-Decomposition, there's a need to ``redefine'' the notion of a ``Neutrosophic SuperHyperTree-Decomposition'' and a ``Neutrosophic SuperHyperTree-Decomposition ''. 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 SuperHyperTree-Decomposition  are redefined to a ``Neutrosophic SuperHyperTree-Decomposition'' 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 SuperHyperTree-Decomposition  more understandable. Assume a Neutrosophic SuperHyperGraph. There are some Neutrosophic SuperHyperClasses if the intended Table holds. Thus  SuperHyperPath,  SuperHyperTree-Decomposition, SuperHyperStar, SuperHyperBipartite, SuperHyperMultiPartite, and   SuperHyperWheel, are
``Neutrosophic SuperHyperPath'', ``Neutrosophic SuperHyperTree-Decomposition'', ``Neutrosophic SuperHyperStar'', ``Neutrosophic SuperHyperBipartite'', ``Neutrosophic SuperHyperMultiPartite'', and ``Neutrosophic SuperHyperWheel'' if the intended Table holds.  A SuperHyperGraph has a ``Neutrosophic SuperHyperTree-Decomposition'' where it's the strongest [the maximum Neutrosophic value from all the SuperHyperTree-Decomposition  amid the maximum value amid all SuperHyperVertices from a SuperHyperTree-Decomposition .] SuperHyperTree-Decomposition . 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 SuperHyperTree-Decomposition 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(-/SuperHyperTree-Decomposition, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel). The aim is to find either the longest SuperHyperTree-Decomposition  or the strongest SuperHyperTree-Decomposition  in those Neutrosophic SuperHyperModels. For the longest SuperHyperTree-Decomposition, called SuperHyperTree-Decomposition, and the strongest SuperHyperTree-Decomposition, called Neutrosophic SuperHyperTree-Decomposition, 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 SuperHyperTree-Decomposition. There isn't any formation of any SuperHyperTree-Decomposition but literarily, it's the deformation of any SuperHyperTree-Decomposition. It, literarily, deforms and it doesn't form.  A basic familiarity with Neutrosophic  SuperHyperTree-Decomposition theory, SuperHyperGraphs, and Neutrosophic SuperHyperGraphs theory are proposed.
\\ \vspace{4mm}
\textbf{Keywords:} Neutrosophic SuperHyperGraph, SuperHyperTree-Decomposition, Cancer's Recognition
   \\
\textbf{AMS Subject Classification:} 05C17, 05C22, 05E45
\section{Applied Notions Under The Scrutiny Of The Motivation Of This Scientific Research}
In this scientific 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 ``Extreme 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 ``Extreme 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 Extreme 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 ``  SuperHyperTree-Decomposition'' 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 Extreme 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 Extreme SuperHyperPath (-/SuperHyperTree-Decomposition, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel). The aim is to find either the optimal   SuperHyperTree-Decomposition or the Extreme   SuperHyperTree-Decomposition in those Extreme SuperHyperModels. Some general results are introduced. Beyond that in SuperHyperStar, all possible Extreme 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 SuperHyperTree-Decomposition. There isn't any formation of any SuperHyperTree-Decomposition but literarily, it's the deformation of any SuperHyperTree-Decomposition. 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   SuperHyperTree-Decomposition'' 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   SuperHyperTree-Decomposition'' 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 ``  SuperHyperTree-Decomposition'' and ``Extreme   SuperHyperTree-Decomposition'' on ``SuperHyperGraph'' and ``Extreme 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 Extreme 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,   SuperHyperTree-Decomposition and Extreme   SuperHyperTree-Decomposition, are figured out in sections ``  SuperHyperTree-Decomposition'' and ``Extreme   SuperHyperTree-Decomposition''. In the sense of tackling on getting results and in Tree-Decomposition to make sense about continuing the research, the ideas of SuperHyperUniform and Extreme 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 Extreme 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 Extreme SuperHyperGraph, in the sections ``Results on SuperHyperClasses'' and ``Results on Extreme 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'',
``  SuperHyperTree-Decomposition'', ``Extreme   SuperHyperTree-Decomposition'', ``Results on SuperHyperClasses'' and ``Results on Extreme 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, ``  SuperHyperTree-Decomposition''. 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{Extreme Preliminaries Of This Scientific Research On the Redeemed Ways}
In this section, the basic material in this scientific research, is referred to [Single Valued Neutrosophic Set](\textbf{Ref.}\cite{HG38},Definition 2.2,p.2), [Neutrosophic Set](\textbf{Ref.}\cite{HG38},Definition 2.1,p.1), [Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.5,p.2), [Characterization of the Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.7,p.3), [t-norm](\textbf{Ref.}\cite{HG38}, Definition 2.7, p.3), and [Characterization of the Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.7,p.3), [Neutrosophic Strength of the Neutrosophic SuperHyperPaths] (\textbf{Ref.}\cite{HG38},Definition 5.3,p.7), and [Different Neutrosophic Types of Neutrosophic SuperHyperEdges (NSHE)] (\textbf{Ref.}\cite{HG38},Definition 5.4,p.7). Also, the new ideas and their clarifications are addressed to \textbf{Ref.}\cite{HG38}.
\\
In this subsection, the basic material which is used in this scientific research, is presented. Also, the new ideas and their clarifications are elicited.
\begin{definition}[Neutrosophic Set](\textbf{Ref.}\cite{HG38},Definition 2.1,p.1).\\
 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{HG38},Definition 2.2,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{HG38},Definition 2.5,p.2).\\
Assume $V'$ is a given set. a \textbf{Neutrosophic SuperHyperGraph} (NSHG) $S$
is a 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{HG38},Definition 2.7,p.3).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a 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{HG38}, Definition 2.7, p.3).\\
 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 a 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{HG38},Definition 2.7,p.3).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a 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 SuperHyperUniform, the some SuperHyperClasses are introduced. It makes to have SuperHyperUniform 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 a 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  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  SuperHyperPaths).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ a 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 Strength of the Neutrosophic SuperHyperPaths](\textbf{Ref.}\cite{HG38},Definition 5.3,p.7).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ A 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)
 $$V_1,E_1,V_2,E_2,V_3,\ldots,V_{s-1},E_{s-1},V_s,$$
have
\begin{itemize}
 \item[$(i)$]\textbf{Neutrosophic t-strength} $(\min\{T(V_i)\},m,n)_{i=1}^s$;
\item[$(ii)$] \textbf{Neutrosophic i-strength} $(m,\min\{I(V_i)\},n)_{i=1}^s;$
\item[$(iii)$] \textbf{Neutrosophic f-strength} $(m,n,\min\{F(V_i)\})_{i=1}^s;$
\item[$(iv)$] \textbf{Neutrosophic strength} $(\min\{T(V_i)\},\min\{I(V_i)\},\min\{F(V_i)\})_{i=1}^s.$
\end{itemize}
\end{definition}

\begin{definition}[Different Neutrosophic Types of Neutrosophic SuperHyperEdges (NSHE)](\textbf{Ref.}\cite{HG38},Definition 5.4,p.7).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a  pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
  \item[$(ix)$] \textbf{Neutrosophic t-connective} if $T(E)\geq$ maximum number of Neutrosophic t-strength of SuperHyperPath (NSHP) from Neutrosophic SuperHyperVertex (NSHV) $V_i$ to Neutrosophic SuperHyperVertex (NSHV) $V_j$ where $1\leq i,j\leq s;$
\item[$(x)$] \textbf{Neutrosophic i-connective} if $I(E)\geq$ maximum number of Neutrosophic i-strength of SuperHyperPath (NSHP) from Neutrosophic SuperHyperVertex (NSHV) $V_i$ to Neutrosophic SuperHyperVertex (NSHV) $V_j$ where $1\leq i,j\leq s;$
\item[$(xi)$] \textbf{Neutrosophic f-connective} if $F(E)\geq$ maximum number of Neutrosophic f-strength of SuperHyperPath (NSHP) from Neutrosophic SuperHyperVertex (NSHV) $V_i$ to Neutrosophic SuperHyperVertex (NSHV) $V_j$ where $1\leq i,j\leq s;$
\item[$(xii)$] \textbf{Neutrosophic connective} if $(T(E),I(E),F(E))\geq$ maximum number of Neutrosophic strength of SuperHyperPath (NSHP) from Neutrosophic SuperHyperVertex (NSHV) $V_i$ to Neutrosophic SuperHyperVertex (NSHV) $V_j$ where $1\leq i,j\leq s.$
\end{itemize}
\end{definition}

 

\begin{definition}(Different Neutrosophic Types of Neutrosophic SuperHyperTree-Decomposition).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called
\begin{itemize}
  \item[$(i)$] \textbf{Neutrosophic e-SuperHyperTree-Decomposition} if it's an ordered pair $(T,\mathcal{T})$ where $T$ is a tree and $\mathcal{T}$ is a family of subtrees of $T$ such that  $T_a\cap T_b\neq \empty$ if $ab\in E;$
\item[$(ii)$] \textbf{Neutrosophic re-SuperHyperTree-Decomposition} if it's an ordered pair $(T,\mathcal{T})$ where $T$ is a tree and $\mathcal{T}$ is a family of subtrees of $T$ such that  $T_a\cap T_b\neq \empty$ if $ab\in E;$ and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$
 \item[$(iii)$] \textbf{Neutrosophic v-SuperHyperTree-Decomposition} if it's an ordered pair $(T,\mathcal{T})$ where $T$ is a tree and $\mathcal{T}$ is a family of subtrees of $T$ such that  $T_a\cap T_b\neq \empty$ if $ab\in E.$
\item[$(iv)$]  \textbf{Neutrosophic rv-SuperHyperTree-Decomposition}  if it's an ordered pair $(T,\mathcal{T})$ where $T$ is a tree and $\mathcal{T}$ is a family of subtrees of $T$ such that  $T_a\cap T_b\neq \empty$ if $ab\in E;$    and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(v)$] \textbf{Neutrosophic SuperHyperTree-Decomposition} if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition.
\end{itemize}
\end{definition}

\begin{definition}((Neutrosophic) SuperHyperTree-Decomposition).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
 \item[$(i)$]
an \textbf{Extreme SuperHyperTree-Decomposition} if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTree-Decomposition;
 \item[$(ii)$]
a \textbf{Neutrosophic SuperHyperTree-Decomposition} if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ for a  Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality  consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperTree-Decomposition;
 \item[$(iii)$]
an \textbf{Extreme SuperHyperTree-Decomposition SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTree-Decomposition; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(iv)$]
a \textbf{Neutrosophic SuperHyperTree-Decomposition SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ 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 the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperTree-Decomposition; and the Neutrosophic power is corresponded to its Neutrosophic coefficient;
\item[$(v)$]
an \textbf{Extreme V-SuperHyperTree-Decomposition} if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTree-Decomposition;
 \item[$(vi)$]
a \textbf{Neutrosophic V-SuperHyperTree-Decomposition} if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperTree-Decomposition;
 \item[$(vii)$]
an \textbf{Extreme V-SuperHyperTree-Decomposition SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTree-Decomposition; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(viii)$]
a \textbf{Neutrosophic SuperHyperTree-Decomposition SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperTree-Decomposition, Neutrosophic re-SuperHyperTree-Decomposition, Neutrosophic v-SuperHyperTree-Decomposition, and Neutrosophic rv-SuperHyperTree-Decomposition and $\mathcal{C}(NSHG)$ 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 the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperTree-Decomposition; and the Neutrosophic power is corresponded to its Neutrosophic coefficient.
\end{itemize}
\end{definition}

\begin{definition}((Extreme/Neutrosophic)$\delta-$SuperHyperTree-Decomposition).\\
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$  Then
\begin{itemize}
 \item[$(i)$] an \textbf{$\delta-$SuperHyperTree-Decomposition} is a Neutrosophic kind of Neutrosophic SuperHyperTree-Decomposition 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{136EQN1}
 \\&& |S\cap N(s)| < |S\cap (V\setminus N(s))|+\delta. \label{136EQN2}
\end{eqnarray*}
The Expression \eqref{136EQN1}, holds if $S$ is an \textbf{$\delta-$SuperHyperOffensive}. And the Expression \eqref{136EQN2}, holds if $S$ is an \textbf{$\delta-$SuperHyperDefensive};
 \item[$(ii)$] a \textbf{Neutrosophic $\delta-$SuperHyperTree-Decomposition} is a Neutrosophic kind of Neutrosophic SuperHyperTree-Decomposition such that either of the following Neutrosophic 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{136EQN3}
 \\&& |S\cap N(s)|_{Neutrosophic} < |S\cap (V\setminus N(s))|_{Neutrosophic}+\delta. \label{136EQN4}
\end{eqnarray*}
The Expression \eqref{136EQN3}, holds if $S$ is a \textbf{Neutrosophic $\delta-$SuperHyperOffensive}. And the Expression \eqref{136EQN4}, holds if $S$ is a \textbf{Neutrosophic $\delta-$SuperHyperDefensive}.
\end{itemize}
\end{definition}
For the sake of having a Neutrosophic SuperHyperTree-Decomposition, 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{136DEF1}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$  It's redefined \textbf{Neutrosophic SuperHyperGraph} if the Table \eqref{136TBL3} holds.
 \begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph Mentioned in the Definition \eqref{136DEF1}}
\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{136TBL3}
\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{136DEF2}
 Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ There are some \textbf{Neutrosophic SuperHyperClasses} if the Table \eqref{136TBL4} holds. Thus  Neutrosophic SuperHyperPath ,  SuperHyperTree-Decomposition, SuperHyperStar, SuperHyperBipartite, SuperHyperMultiPartite, and   SuperHyperWheel, are
 \textbf{Neutrosophic SuperHyperPath},  \textbf{Neutrosophic SuperHyperCycle}, \textbf{Neutrosophic SuperHyperStar}, \textbf{Neutrosophic SuperHyperBipartite}, \textbf{Neutrosophic SuperHyperMultiPartite}, and \textbf{Neutrosophic SuperHyperWheel} if the Table \eqref{136TBL4} holds.
 \begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph, Mentioned in the Definition \eqref{136DEF2}}
\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{136TBL4}
\end{table}
\end{definition}
It's useful to define a ``Neutrosophic'' version of a Neutrosophic SuperHyperTree-Decomposition. Since there's more ways to get type-results to make a Neutrosophic SuperHyperTree-Decomposition more Neutrosophicly understandable.
\\
For the sake of having a Neutrosophic SuperHyperTree-Decomposition, there's a need to ``\textbf{redefine}'' the Neutrosophic notion of ``Neutrosophic SuperHyperTree-Decomposition''. 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{136DEF1}
 Assume a SuperHyperTree-Decomposition. It's redefined a \textbf{Neutrosophic SuperHyperTree-Decomposition} if the Table \eqref{136TBL1} holds.  \begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph Mentioned in the Definition \eqref{136DEF1}}
\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{136TBL1}
\end{table}
\end{definition}
\section{
Extreme SuperHyperTree-Decomposition But As The Extensions Excerpt From Dense And Super Forms}
\begin{example}\label{136EXM1}
 Assume a Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Extreme Figures in every Extreme items.
 \begin{itemize}
 \item On the Figure \eqref{136NSHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.  $E_1$ and $E_3$ are some empty Extreme  
 SuperHyperEdges but $E_2$ is a loop Extreme SuperHyperEdge and $E_4$ is a 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 a Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Extreme SuperHyperTree-Decomposition.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}=\{E_2,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}=\{V_i\}_{i\neq2,3}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}=2z^2.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG1.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG1}
\end{figure}
 \item On the Figure \eqref{136NSHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.  $E_1,E_2$ and $E_3$ are some empty Extreme SuperHyperEdges but $E_4$ is a 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 a Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Extreme SuperHyperTree-Decomposition.
\begin{eqnarray*}
 &&
    \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}=\{V_i\}_{i\neq2,3}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}=3z^2.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG2.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG2}
 \end{figure}
  \item On the Figure \eqref{136NSHG3}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
    \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}=\{V_i\}_{i=1,3}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}=3z^2.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG3.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG3}
 \end{figure}
 \item On the Figure \eqref{136NSHG4}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
  \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}=\{E_4,E_5,E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}=2z^4.
   \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}=\{V_1,V_2,V_3,V_4,H\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
   \\&&
  =(\text{Five Choose Three})\times2z^5.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG4.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG4}
 \end{figure}
  \item On the Figure \eqref{136NSHG5}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}=\{E_1,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
 \\&&
 =(\text{Four Choose Two})z^2.
 \\&&
     \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}=\{V_{13},V_5,V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
  \\&&
 =(2\time2\times5\times4+4\times4+5\times5)z^3.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG5.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG5}
 \end{figure}
  \item On the Figure \eqref{136NSHG6}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}
  \\&&   =\{E_i,E_j,E_{22}\}_{i=1,j=23}^{10,32}.
           \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
  \\&& =22z^{21}.
    \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
  \\&&
  =\{V_i\}_{i=1}^{22}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
  \\&& =22z^{22}.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG6.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG6}
 \end{figure}
 \item On the Figure \eqref{136NSHG7},  the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}=
           \\&&
           \{E_{i}\}_{i\neq11}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
 \\&&
=11z^{14}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}=
    \\&&
           \{V_i\}_{i=1}^{14}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
 \\&&
=11z^{14}.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG7.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG7}
 \end{figure}
 \item On the Figure \eqref{136NSHG8}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}=
           \\&&
           \{E_3,E_4,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
 \\&&
=3\times2z^3.
\\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}=
    \\&&
           \{V_8,V_{14},V_{13},V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
 \\&&
=11\times8z^4.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG8.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG8}
 \end{figure}
 \item On the Figure \eqref{136NSHG9}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}
  \\&&   =\{E_i,E_{22},E_{23}\}_{i=1}^{10}.
           \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
  \\&& =11z^{11}.
    \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
  \\&&   =\{V_{i},V_{21},V_{22}\}_{i=1}^{10}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
  \\&& =11z^{12}.
    \end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG9.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG9}
 \end{figure}
 \item On the Figure \eqref{136NSHG10}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}=
           \\&&
           \{E_2,E_6,E_4,E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
  \\&&
 =3\times2z^4.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}=
    \\&&
         =\{V_6,V_{13},V_{12},V_{14},V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
 \\&&
(4\times4+2\times3\times4)z^5.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG10.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG10}
 \end{figure}
  \item On the Figure \eqref{136NSHG11}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}=
     \\&&      
          \{E_1,E_6,E_5,E_4,E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
 \\&&   
 =3z^5.
  \\&&
    \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}=\{V_1,V_5,V_6,V_4,V_2\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
 \\&&
 =3z^5.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG11.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG11}
 \end{figure}
 \item On the Figure \eqref{136NSHG12}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}=\{E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
   \\&&
 =5z^2.
 \\&&
    \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
  \\&&
   =\{V_8,V_1,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}=
  \\&&
=5\times4z^3.
\end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG12.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG12}
 \end{figure}
  \item On the Figure \eqref{136NSHG13}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}=
     \\&&      
          \{E_1,E_{10},E_8,E_4,E_2,E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
 \\&&   
 =3\times2z^6.
  \\&&
    \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}=\{V_2,V_3,V_6,V_4,V_5,V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
 \\&&
 =3\times2z^6.
 \end{eqnarray*}
  \begin{figure}
 \includegraphics[width=100mm]{136NSHG13.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG13}
 \end{figure}
  \item On the Figure \eqref{136NSHG14}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}=\{E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}=\{V_2,V_1,V_3\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}=
z^3.
 \end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG14.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG14}
 \end{figure}
  \item On the Figure \eqref{136NSHG15}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}
   \\&&        
          =\{E_i\}_{i=1}^5.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}=2z^5.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
   \\&&        
          =\{V_i\}_{i=1}^6.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
  =2z^6.
  \end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG15.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG15}
 \end{figure}
  \item On the Figure \eqref{136NSHG16}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}
    \\&&       
          =\{E_i\}_{i=1}^5.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}=2z^5.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
    \\&&
   =\{V_1,V_2,V_6,V_{10},V_{17},V_{22}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
    \\&&
  =4\times2\times5z^6.
  \end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG16.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG16}
 \end{figure}
  \item On the Figure \eqref{136NSHG17}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}
    \\&&       
          =\{E_i\}_{i=1}^6.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}=2z^6.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
    \\&&
   =\{V_{27},V_1,V_2,V_6,V_{10},V_{17},V_{22}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
    \\&&
  =7\times4\times2\times5z^7.
  \end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG17.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG17}
 \end{figure}
  \item On the Figure \eqref{136NSHG18}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}
    \\&&       
          =\{E_i\}_{i=1}^6.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}=2z^6.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
    \\&&
   =\{V_{27},V_1,V_2,V_6,V_{10},V_{17},V_{22}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
    \\&&
  =2\times6\times4\times2\times5z^7.
  \end{eqnarray*}
   \begin{figure}
 \includegraphics[width=100mm]{136NSHG18.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG18}
 \end{figure}
  \item On the Figure \eqref{136NSHG19}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}
    \\&&       
          =\{E_i\}_{i=1}^{11}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}=12z^{11}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
    \\&&
  =\{V_i\}_{i=1}^{11}.
   \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
    \\&&
  =11z^{11}.
  \end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG19.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG19}
 \end{figure}
  \item On the Figure \eqref{136NSHG20}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}
     \\&&      
          =\{E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
  \\&&
 =\text{Ten Choose Two}z^2.
 \\&&
    \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
     \\&&
    =\{V,V_1,V'\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
  \\&&
=az^3.
 \end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{136NSHG20.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{136NSHG20}
 \end{figure}
 \item On the Figure \eqref{95NHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}
     \\&&      
          =\{E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
  \\&&
 =z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
     \\&&
    =\{V_2,V_1,V_3\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
  \\&&
 =8z^3.
 \end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{95NHG1.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{95NHG1}
 \end{figure}
 \item On the Figure \eqref{95NHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTree-Decomposition, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}
     \\&&      
          =\{E_i\}_{i=1}^5.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
  \\&&
=2z^5.
 \\&&
    \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
     \\&&
   =\{M,V_1,V_3,V_{10},V_6,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
  \\&&
=8\times3\times5z^6.
  \end{eqnarray*}
 \begin{figure}
 \includegraphics[width=100mm]{95NHG2.png}  
   \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM1} }
 \label{95NHG2}
 \end{figure}
\end{itemize}
\end{example}

\section{The Extreme Departures on The Theoretical Results Toward Theoretical Motivations}
The previous Extreme approach apply on the upcoming Extreme results on Extreme SuperHyperClasses.
\begin{proposition}
Assume a connected Extreme SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}
     \\&&      
          =\{E_i\}_{i=1}^{|E_{NSHG}|}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
  \\&&
 =2z^{|E_{NSHG}|}.
 \\&&  
    \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
     \\&&
             =\{V^{EXTERNAL}_i\}_{i=1}^{|E_{NSHG}|}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
  \\&&
 =az^{|E_{NSHG}|}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
  \\&&\ldots,
   \\&&V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}},E_{\frac{|E_{NSHG}|}{3}}
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,V^{EXTERNAL}_2,
  \\&&\ldots,
   \\&&E_{\frac{|E_{NSHG}|}{3}},V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}}
 \end{eqnarray*}
be a longest path taken from a connected Extreme SuperHyperPath $ESHP:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTree-Decomposition. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM18a}
In the Figure \eqref{136NSHG18a}, the connected Extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Extreme SuperHyperSet,  in the Extreme SuperHyperModel \eqref{136NSHG18a}, is the  SuperHyperTree-Decomposition.
\begin{figure}
\includegraphics[width=100mm]{136NSHG18.png}
\caption{a Extreme SuperHyperPath Associated to the Notions of  Extreme SuperHyperTree-Decomposition in the Example \eqref{136EXM18a}}
\label{136NSHG18a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}
     \\&&      
          =\{E_i\}_{i=1}^{|E_{NSHG}|}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
  \\&&
 =(|E_{NSHG}|-1)z^{|E_{NSHG}|-1}.
 \\&&  
    \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
     \\&&
             =\{V^{EXTERNAL}_i\}_{i=1}^{|E_{NSHG}|-1}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
  \\&&
 =az^{|E_{NSHG}|-1}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
  \\&&\ldots,
   \\&&V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}},E_{\frac{|E_{NSHG}|}{3}}
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,V^{EXTERNAL}_2,
  \\&&\ldots,
   \\&&E_{\frac{|E_{NSHG}|}{3}},V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}}
 \end{eqnarray*}
  be a longest path taken from a connected Extreme SuperHyperCycle $ESHC:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTree-Decomposition. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM19a}
In the Figure \eqref{136NSHG19a}, the connected Extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Extreme SuperHyperSet, in the Extreme SuperHyperModel \eqref{136NSHG19a},
is the Extreme  SuperHyperTree-Decomposition.
\begin{figure}
\includegraphics[width=100mm]{136NSHG19.png}
\caption{a Extreme SuperHyperCycle Associated to the Extreme Notions of  Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM19a}}
\label{136NSHG19a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperStar $ESHS:(V,E).$ Then  
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}=\{E,E'\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
  \\&&
 =(|E_{NSHG}|\text{Choose Two})z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
    \\&& =\{V,CENTER,V'\}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
  \\&&
 =(|E_{NSHG}|\text{Choose Two})z^3.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&CENTER,E_2
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,CENTER
 \end{eqnarray*}
be a longest path taken a connected Extreme SuperHyperStar $ESHS:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTree-Decomposition. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM20a}
In the Figure \eqref{136NSHG20a}, 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{136NSHG20a}, is the  Extreme SuperHyperTree-Decomposition.
\begin{figure}
\includegraphics[width=100mm]{136NSHG20.png}
\caption{a Extreme SuperHyperStar Associated to the Extreme Notions of  Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM20a}}
\label{136NSHG20a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}
         \\&&  
          =\{{E_i,E_j}_{E_i\in P_i,~E_j\in P_j,~i\neq j}\}_{i,j=1}^{i+j=|P^{\min}_{NSHG}|}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
  \\&&
 =(|P^{i}_{NSHG}|\times|P^{j}_{NSHG}|)z^{|P^{\min}_{NSHG}|}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
    \\&&
          =\{{V^{EXTERNAL},V^{EXTERNAL}}_{V^{EXTERNAL}\in P_i,~V^{EXTERNAL}_j\in P_j,~i\neq j}\}_{i,j=1}^{i+j=|P^{\min}_{NSHG}|}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
  \\&&
 =(|P^{i}_{NSHG}|\times|P^{j}_{NSHG}|)z^{|P^{\min}_{NSHG}|}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
  \\&&\ldots,
  \\&&V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|}.
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,V^{EXTERNAL}_2,
  \\&&\ldots,
  \\&&E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|}
 \end{eqnarray*}
  is a longest path taken from a connected   Extreme SuperHyperBipartite $ESHB:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTree-Decomposition. The latter is straightforward. Then there's no at least one SuperHyperTree-Decomposition. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperTree-Decomposition could be applied. There are only two SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest SuperHyperTree-Decomposition taken from a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-of-SuperHyperPart SuperHyperEdges are attained in any solution
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM21a}
In the Extreme Figure \eqref{136NSHG21a}, 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{136NSHG21a}, is the Extreme  SuperHyperTree-Decomposition.
\begin{figure}
\includegraphics[width=100mm]{136NSHG21.png}
\caption{Extreme SuperHyperBipartite Extreme Associated to the Extreme Notions of  Extreme SuperHyperTree-Decomposition in the Example \eqref{136EXM21a}}
\label{136NSHG21a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}
         \\&&  
          =\{{E_i,E_j}_{E_i\in P_i,~E_j\in P_j,~i\neq j}\}_{i,j=1}^{i+j=|P^{\min}_{NSHG}|}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
  \\&&
 =(\prod_{i,j}|P^{i}_{NSHG}|\times|P^{j}_{NSHG}|)z^{|P^{\min}_{NSHG}|}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
    \\&&
          =\{{V^{EXTERNAL},V^{EXTERNAL}}_{V^{EXTERNAL}\in P_i,~V^{EXTERNAL}_j\in P_j,~i\neq j}\}_{i,j=1}^{i+j=|P^{\min}_{NSHG}|}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
  \\&&
 =(\prod_{i,j}|P^{i}_{NSHG}|\times|P^{j}_{NSHG}|)z^{|P^{\min}_{NSHG}|}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
  \\&&\ldots,
  \\&&V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|}.
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E_1,V^{EXTERNAL}_1,
 \\&&E_2,V^{EXTERNAL}_2,
  \\&&\ldots,
  \\&&E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|}
 \end{eqnarray*}
 is a longest SuperHyperTree-Decomposition taken from a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTree-Decomposition. The latter is straightforward.  Then there's no at least one SuperHyperTree-Decomposition. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperTree-Decomposition could be applied. There are only $z'$ SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the  
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-of-SuperHyperPart SuperHyperEdges are attained in any solution
    \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, 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{136NSHG22a}, is the  Extreme SuperHyperTree-Decomposition.
\begin{figure}
\includegraphics[width=100mm]{136NSHG22.png}
\caption{a Extreme  SuperHyperMultipartite Associated to the Notions of  Extreme SuperHyperTree-Decomposition in the Example \eqref{136EXM22a}}
\label{136NSHG22a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition}}=\{E_i\}_{i=1}^{|E^{*}_{NSHG}|}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Tree-Decomposition SuperHyperPolynomial}}
  \\&&
 =|E^{*}_{NSHG}|\times|E_{NSHG}|z^{|E^{*}_{NSHG}|}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme V-Tree-Decomposition}}
    \\&&
   = \{V^{EXTERNAL}\in E_i\}_{i=1}^{|E^{*}_{NSHG}|}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme V-Tree-Decomposition SuperHyperPolynomial}}}
  \\&&
 =|E^{*}_{NSHG}|\times|E_{NSHG}|z^{|E^{*}_{NSHG}|}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E^{*}_1,
 \\&&CENTER,E^{*}_2
 \end{eqnarray*}
  \begin{eqnarray*}
 && P:
 \\&&
 E^{*}_1,V^{EXTERNAL}_1,
 \\&&E^{*}_2,CENTER
 \end{eqnarray*}
  is a longest SuperHyperTree-Decomposition taken from a connected Extreme SuperHyperWheel $ESHW:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTree-Decomposition. The latter is straightforward. Then there's at least one SuperHyperTree-Decomposition. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperTree-Decomposition could be applied. The unique embedded SuperHyperTree-Decomposition proposes some longest SuperHyperTree-Decomposition excerpt from some representatives. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM23a}
In the Extreme Figure \eqref{136NSHG23a}, 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{136NSHG23a}, is the  Extreme SuperHyperTree-Decomposition.
\begin{figure}
\includegraphics[width=100mm]{136NSHG23.png}
\caption{a Extreme SuperHyperWheel Extreme Associated to the Extreme Notions of  Extreme SuperHyperTree-Decomposition in the Extreme Example \eqref{136EXM23a}}
\label{136NSHG23a}
\end{figure}
            \end{example}
\section{The Surveys of Mathematical Sets On The Results But As The Initial Motivation}
For the SuperHyperTree-Decomposition,  Extreme SuperHyperTree-Decomposition, and the Extreme SuperHyperTree-Decomposition, some general results are introduced.
\begin{remark}
 Let remind that the Extreme SuperHyperTree-Decomposition is ``redefined'' on the positions of the alphabets.
\end{remark}
\begin{corollary}
 Assume Extreme SuperHyperTree-Decomposition. Then
 \begin{eqnarray*}
&& Extreme ~SuperHyperTree-Decomposition=\\&&\{the   SuperHyperTree-Decomposition of the SuperHyperVertices ~|~\\&&\max|SuperHyperOffensive \\&&SuperHyperTree-Decomposition
\\&&
|_{Extreme cardinality amid those SuperHyperTree-Decomposition.}\}
  \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 Extreme SuperHyperGraph on the same identical letter of the alphabet. Then the notion of Extreme SuperHyperTree-Decomposition and SuperHyperTree-Decomposition coincide.
\end{corollary}
\begin{corollary}
Assume a Extreme SuperHyperGraph on the same identical letter of the alphabet. Then a consecutive sequence of the SuperHyperVertices is a Extreme SuperHyperTree-Decomposition if and only if it's a SuperHyperTree-Decomposition.
\end{corollary}
\begin{corollary}
Assume a Extreme SuperHyperGraph on the same identical letter of the alphabet. Then a consecutive sequence of the SuperHyperVertices is a strongest SuperHyperTree-Decomposition if and only if it's a longest SuperHyperTree-Decomposition.
\end{corollary}
\begin{corollary}
Assume SuperHyperClasses of a Extreme SuperHyperGraph on the same identical letter of the alphabet. Then its Extreme SuperHyperTree-Decomposition is its SuperHyperTree-Decomposition and reversely.
\end{corollary}
\begin{corollary}
Assume a Extreme SuperHyperPath(-/SuperHyperTree-Decomposition, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel) on the same identical letter of the alphabet. Then its Extreme SuperHyperTree-Decomposition is its SuperHyperTree-Decomposition and reversely.
\end{corollary}
\begin{corollary}
 Assume a Extreme SuperHyperGraph. Then its Extreme SuperHyperTree-Decomposition isn't well-defined if and only if its SuperHyperTree-Decomposition isn't well-defined.
\end{corollary}
\begin{corollary}
 Assume SuperHyperClasses of a Extreme SuperHyperGraph. Then its Extreme SuperHyperTree-Decomposition isn't well-defined if and only if its SuperHyperTree-Decomposition isn't well-defined.
\end{corollary}
\begin{corollary}
Assume a Extreme SuperHyperPath(-/SuperHyperTree-Decomposition, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel). Then its Extreme SuperHyperTree-Decomposition isn't well-defined if and only if its SuperHyperTree-Decomposition isn't well-defined.
\end{corollary}
\begin{corollary}
 Assume a Extreme SuperHyperGraph. Then its Extreme SuperHyperTree-Decomposition is well-defined if and only if its SuperHyperTree-Decomposition is well-defined.
\end{corollary}
\begin{corollary}
 Assume SuperHyperClasses of a Extreme SuperHyperGraph. Then its Extreme SuperHyperTree-Decomposition is well-defined if and only if its SuperHyperTree-Decomposition is well-defined.
\end{corollary}
\begin{corollary}
Assume a Extreme SuperHyperPath(-/SuperHyperTree-Decomposition, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel). Then its Extreme SuperHyperTree-Decomposition is well-defined if and only if its SuperHyperTree-Decomposition is well-defined.
\end{corollary}

%

\begin{proposition}
Let $ESHG:(V,E)$  be a Extreme SuperHyperGraph. Then $V$ is
\begin{itemize}
 \item[$(i):$]  the dual SuperHyperDefensive SuperHyperTree-Decomposition;
  \item[$(ii):$]  the strong dual SuperHyperDefensive SuperHyperTree-Decomposition;
    \item[$(iii):$]  the connected dual SuperHyperDefensive SuperHyperTree-Decomposition;
      \item[$(iv):$]  the $\delta$-dual SuperHyperDefensive SuperHyperTree-Decomposition;
        \item[$(v):$]  the strong $\delta$-dual SuperHyperDefensive SuperHyperTree-Decomposition;
          \item[$(vi):$]  the connected $\delta$-dual SuperHyperDefensive SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $NTG:(V,E,\sigma,\mu)$  be a Extreme SuperHyperGraph. Then $\emptyset$ is
\begin{itemize}
 \item[$(i):$] the SuperHyperDefensive SuperHyperTree-Decomposition;
  \item[$(ii):$]  the strong SuperHyperDefensive SuperHyperTree-Decomposition;
    \item[$(iii):$]  the connected defensive SuperHyperDefensive SuperHyperTree-Decomposition;
      \item[$(iv):$]  the $\delta$-SuperHyperDefensive SuperHyperTree-Decomposition;
        \item[$(v):$]  the strong $\delta$-SuperHyperDefensive SuperHyperTree-Decomposition;
          \item[$(vi):$]  the connected $\delta$-SuperHyperDefensive SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a Extreme SuperHyperGraph. Then an independent SuperHyperSet is
\begin{itemize}
 \item[$(i):$] the SuperHyperDefensive SuperHyperTree-Decomposition;
  \item[$(ii):$]  the strong SuperHyperDefensive SuperHyperTree-Decomposition;
    \item[$(iii):$]  the connected SuperHyperDefensive SuperHyperTree-Decomposition;
      \item[$(iv):$]  the $\delta$-SuperHyperDefensive SuperHyperTree-Decomposition;
        \item[$(v):$]  the strong $\delta$-SuperHyperDefensive SuperHyperTree-Decomposition;
          \item[$(vi):$]  the connected $\delta$-SuperHyperDefensive SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}

\begin{proposition}
Let $ESHG:(V,E)$ be a Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyperTree-Decomposition/SuperHyperPath. Then $V$ is a maximal
\begin{itemize}
 \item[$(i):$] SuperHyperDefensive SuperHyperTree-Decomposition;
  \item[$(ii):$]  strong SuperHyperDefensive SuperHyperTree-Decomposition;
    \item[$(iii):$]  connected SuperHyperDefensive SuperHyperTree-Decomposition;
      \item[$(iv):$]  $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyperTree-Decomposition;
        \item[$(v):$]  strong $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyperTree-Decomposition;
          \item[$(vi):$]  connected $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyperTree-Decomposition;
\end{itemize}
Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a Extreme SuperHyperGraph which is a  SuperHyperUniform SuperHyperWheel. Then $V$ is a maximal
\begin{itemize}
 \item[$(i):$] dual SuperHyperDefensive SuperHyperTree-Decomposition;
  \item[$(ii):$]  strong dual SuperHyperDefensive SuperHyperTree-Decomposition;
    \item[$(iii):$]  connected dual SuperHyperDefensive SuperHyperTree-Decomposition;
      \item[$(iv):$]  $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyperTree-Decomposition;
        \item[$(v):$]  strong $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyperTree-Decomposition;
          \item[$(vi):$]  connected $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyperTree-Decomposition;
\end{itemize}
Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}

\begin{proposition}
Let $ESHG:(V,E)$ be a Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyperTree-Decomposition/SuperHyperPath. Then the number of
\begin{itemize}
 \item[$(i):$] the SuperHyperTree-Decomposition;
  \item[$(ii):$] the SuperHyperTree-Decomposition;
    \item[$(iii):$]  the connected SuperHyperTree-Decomposition;
      \item[$(iv):$]  the $\mathcal{O}(ESHG)$-SuperHyperTree-Decomposition;
        \item[$(v):$]  the strong $\mathcal{O}(ESHG)$-SuperHyperTree-Decomposition;
          \item[$(vi):$]  the connected $\mathcal{O}(ESHG)$-SuperHyperTree-Decomposition.
\end{itemize}
is one and it's only $V.$ Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyperWheel. Then the number of
\begin{itemize}
 \item[$(i):$] the dual SuperHyperTree-Decomposition;
  \item[$(ii):$] the dual  SuperHyperTree-Decomposition;
    \item[$(iii):$]  the dual connected SuperHyperTree-Decomposition;
      \item[$(iv):$]  the dual $\mathcal{O}(ESHG)$-SuperHyperTree-Decomposition;
        \item[$(v):$]  the strong dual $\mathcal{O}(ESHG)$-SuperHyperTree-Decomposition;
          \item[$(vi):$]  the connected dual $\mathcal{O}(ESHG)$-SuperHyperTree-Decomposition.
\end{itemize}
is one and it's only $V.$ Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a Extreme 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 SuperHyperTree-Decomposition;
  \item[$(ii):$]  strong dual SuperHyperDefensive SuperHyperTree-Decomposition;
    \item[$(iii):$]  connected dual SuperHyperDefensive SuperHyperTree-Decomposition;
      \item[$(iv):$]  $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperTree-Decomposition;
        \item[$(v):$]  strong $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperTree-Decomposition;
          \item[$(vi):$]  connected $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a Extreme 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 SuperHyperTree-Decomposition;
  \item[$(ii):$]  strong  SuperHyperDefensive SuperHyperTree-Decomposition;
    \item[$(iii):$]  connected  SuperHyperDefensive SuperHyperTree-Decomposition;
      \item[$(iv):$]  $\delta$-SuperHyperDefensive SuperHyperTree-Decomposition;
        \item[$(v):$]  strong $\delta$-SuperHyperDefensive SuperHyperTree-Decomposition;
          \item[$(vi):$]  connected $\delta$-SuperHyperDefensive SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}

\begin{proposition}
Let $ESHG:(V,E)$ be a Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyperStar/SuperHyperComplete SuperHyperBipartite/SuperHyperComplete SuperHyperMultipartite. Then Then the number of \begin{itemize}
 \item[$(i):$] dual SuperHyperDefensive SuperHyperTree-Decomposition;
  \item[$(ii):$]  strong dual SuperHyperDefensive SuperHyperTree-Decomposition;
    \item[$(iii):$]  connected dual SuperHyperDefensive SuperHyperTree-Decomposition;
      \item[$(iv):$]  $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperTree-Decomposition;
        \item[$(v):$]  strong $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperTree-Decomposition;
          \item[$(vi):$]  connected $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperTree-Decomposition.
\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{proposition}
Let $ESHG:(V,E)$  be a Extreme SuperHyperGraph. The number of connected component is $|V-S|$ if there's a SuperHyperSet which is a dual
\begin{itemize}
 \item[$(i):$]  SuperHyperDefensive SuperHyperTree-Decomposition;
  \item[$(ii):$]  strong SuperHyperDefensive SuperHyperTree-Decomposition;
    \item[$(iii):$] connected SuperHyperDefensive SuperHyperTree-Decomposition;
      \item[$(iv):$]  SuperHyperTree-Decomposition;
        \item[$(v):$]  strong 1-SuperHyperDefensive SuperHyperTree-Decomposition;
          \item[$(vi):$]  connected 1-SuperHyperDefensive SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$  be a Extreme SuperHyperGraph. Then the number is at most $\mathcal{O}(ESHG)$ and  the Extreme number is at most $\mathcal{O}_n(ESHG).$
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$  be a Extreme SuperHyperGraph which is SuperHyperComplete. The number is $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and  the Extreme 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 SuperHyperTree-Decomposition;
  \item[$(ii):$] strong
  SuperHyperDefensive SuperHyperTree-Decomposition;
    \item[$(iii):$] connected SuperHyperDefensive SuperHyperTree-Decomposition;
      \item[$(iv):$]  $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperTree-Decomposition;
        \item[$(v):$]  strong $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperTree-Decomposition;
          \item[$(vi):$]  connected $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a Extreme SuperHyperGraph which is $\emptyset.$ The number is  $0$ and  the Extreme number is $0,$ for an independent SuperHyperSet in the setting of dual
\begin{itemize}
 \item[$(i):$] SuperHyperDefensive SuperHyperTree-Decomposition;
  \item[$(ii):$]  strong  SuperHyperDefensive SuperHyperTree-Decomposition;
    \item[$(iii):$] connected  SuperHyperDefensive SuperHyperTree-Decomposition;
      \item[$(iv):$]  $0$-SuperHyperDefensive SuperHyperTree-Decomposition;
        \item[$(v):$]  strong $0$-SuperHyperDefensive SuperHyperTree-Decomposition;
          \item[$(vi):$]  connected $0$-SuperHyperDefensive SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}

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

\begin{proposition}
Let $ESHG:(V,E)$ be a Extreme SuperHyperGraph which is SuperHyperTree-Decomposition/SuperHyperPath/SuperHyperWheel. The number is  $\mathcal{O}(ESHG:(V,E))$ and  the Extreme number is $\mathcal{O}_n(ESHG:(V,E)),$ in the setting of a dual
\begin{itemize}
 \item[$(i):$]  SuperHyperDefensive SuperHyperTree-Decomposition;
  \item[$(ii):$]  strong  SuperHyperDefensive SuperHyperTree-Decomposition;
    \item[$(iii):$] connected  SuperHyperDefensive SuperHyperTree-Decomposition;
      \item[$(iv):$]  $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyperTree-Decomposition;
        \item[$(v):$]  strong $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyperTree-Decomposition;
          \item[$(vi):$]  connected $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}

\begin{proposition}
Let $ESHG:(V,E)$  be a Extreme SuperHyperGraph which is SuperHyperStar/complete SuperHyperBipartite/complete SuperHyperMultiPartite. The number is  $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and  the Extreme 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 SuperHyperTree-Decomposition;
  \item[$(ii):$]  strong  SuperHyperDefensive SuperHyperTree-Decomposition;
    \item[$(iii):$] connected  SuperHyperDefensive SuperHyperTree-Decomposition;
      \item[$(iv):$]  $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperTree-Decomposition;
        \item[$(v):$]  strong $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperTree-Decomposition;
          \item[$(vi):$]  connected $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
 Let $\mathcal{NSHF}:(V,E)$ be a SuperHyperFamily of the $ESHGs:(V,E)$ Extreme 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  Extreme SuperHyperGraphs.
\end{proposition}
%
\begin{proposition}
  Let $ESHG:(V,E)$  be a strong Extreme SuperHyperGraph. If $S$ is a dual SuperHyperDefensive SuperHyperTree-Decomposition, 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{proposition}
   Let $ESHG:(V,E)$  be a strong Extreme SuperHyperGraph. If $S$ is a dual SuperHyperDefensive SuperHyperTree-Decomposition, then
    \begin{itemize}
\item[$(i)$]
$S$ is SuperHyperTree-Decomposition set;
\item[$(ii)$]
there's $S\subseteq S'$ such that $|S'|$ is SuperHyperChromatic number.
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $ESHG:(V,E)$  be a strong Extreme SuperHyperGraph. Then
      \begin{itemize}
\item[$(i)$]
$\Gamma\leq\mathcal{O};$
\item[$(ii)$]
$\Gamma_s\leq\mathcal{O}_n.$
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $ESHG:(V,E)$  be a strong Extreme 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{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 SuperHyperTree-Decomposition;
\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 SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\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 SuperHyperTree-Decomposition;
\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 SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $ESHG:(V,E)$  be an even SuperHyperTree-Decomposition. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n}\}$ is  a dual SuperHyperDefensive SuperHyperTree-Decomposition;
\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  SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $ESHG:(V,E)$  be an odd SuperHyperTree-Decomposition. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n-1}\}$ is  a  dual  SuperHyperDefensive SuperHyperTree-Decomposition;
\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 SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $ESHG:(V,E)$  be SuperHyperStar. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{c\}$ is  a dual maximal SuperHyperTree-Decomposition;
\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 SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\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 SuperHyperTree-Decomposition;
\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 SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\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 SuperHyperTree-Decomposition;
\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 SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}

\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 SuperHyperTree-Decomposition;
\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 SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $\mathcal{NSHF}:(V,E)$ be a $m$-SuperHyperFamily of Extreme SuperHyperStars with common Extreme SuperHyperVertex SuperHyperSet. Then
 \begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{c_1,c_2,\cdots,c_m\}$ is  a dual SuperHyperDefensive SuperHyperTree-Decomposition 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  SuperHyperTree-Decomposition for $\mathcal{NSHF}:(V,E).$
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $\mathcal{NSHF}:(V,E)$ be an $m$-SuperHyperFamily of odd SuperHyperComplete SuperHyperGraphs with common Extreme 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 SuperHyperTree-Decomposition 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 SuperHyperTree-Decomposition for $\mathcal{NSHF}:(V,E).$
\end{itemize}
\end{proposition}
\begin{proposition}
  Let $\mathcal{NSHF}:(V,E)$ be a $m$-SuperHyperFamily of even SuperHyperComplete SuperHyperGraphs with common Extreme SuperHyperVertex SuperHyperSet. Then
\begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ is  a dual SuperHyperDefensive SuperHyperTree-Decomposition 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 SuperHyperTree-Decomposition for $\mathcal{NSHF}:(V,E).$
\end{itemize}
\end{proposition}
\begin{proposition}
   Let $ESHG:(V,E)$ be a strong Extreme SuperHyperGraph. Then following statements hold;
\begin{itemize}
 \item[$(i)$] if $s\geq t$ and a SuperHyperSet $S$ of SuperHyperVertices is an t-SuperHyperDefensive SuperHyperTree-Decomposition, then $S$ is an s-SuperHyperDefensive SuperHyperTree-Decomposition;
  \item[$(ii)$] if $s\leq t$ and a SuperHyperSet $S$ of SuperHyperVertices is a dual t-SuperHyperDefensive SuperHyperTree-Decomposition, then $S$ is a dual s-SuperHyperDefensive  SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
   Let $ESHG:(V,E)$  be a strong Extreme SuperHyperGraph. Then following statements hold;
\begin{itemize}
 \item[$(i)$] if $s\geq t+2$ and a SuperHyperSet $S$ of SuperHyperVertices is an t-SuperHyperDefensive  SuperHyperTree-Decomposition, then $S$ is an s-SuperHyperPowerful  SuperHyperTree-Decomposition;
  \item[$(ii)$] if $s\leq t$ and a SuperHyperSet $S$ of SuperHyperVertices is a dual t-SuperHyperDefensive SuperHyperTree-Decomposition, then $S$ is a dual s-SuperHyperPowerful  SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}

\begin{proposition}
   Let $ESHG:(V,E)$  be a[an] [V-]SuperHyperUniform-strong-Extreme 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 SuperHyperTree-Decomposition;
  \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 SuperHyperTree-Decomposition;
   \item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is an V-SuperHyperDefensive  SuperHyperTree-Decomposition;
     \item[$(iv)$]  if $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is a dual V-SuperHyperDefensive SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
   Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Extreme 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  SuperHyperTree-Decomposition;
  \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  SuperHyperTree-Decomposition;
   \item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an V-SuperHyperDefensive SuperHyperTree-Decomposition;
     \item[$(iv)$]  $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is a dual V-SuperHyperDefensive  SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
   Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Extreme 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  SuperHyperTree-Decomposition;
  \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  SuperHyperTree-Decomposition;
   \item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an $(\mathcal{O}-1)$-SuperHyperDefensive  SuperHyperTree-Decomposition;
     \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  SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
   Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Extreme 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  SuperHyperTree-Decomposition;
  \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  SuperHyperTree-Decomposition;
   \item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is $(\mathcal{O}-1)$-SuperHyperDefensive  SuperHyperTree-Decomposition;
     \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  SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}
\begin{proposition}
   Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Extreme SuperHyperGraph which is SuperHyperTree-Decomposition. 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  SuperHyperTree-Decomposition;
  \item[$(ii)$]  $\forall a\in V\setminus S,~|N_s(a)\cap S|> 2$ if $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyperTree-Decomposition;
   \item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an 2-SuperHyperDefensive  SuperHyperTree-Decomposition;
     \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  SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}

\begin{proposition}
   Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Extreme SuperHyperGraph which is SuperHyperTree-Decomposition. Then following statements hold;
\begin{itemize}
 \item[$(i)$] if $\forall a\in S,~|N_s(a)\cap S|< 2,$ then $ESHG:(V,E)$ is an 2-SuperHyperDefensive  SuperHyperTree-Decomposition;
  \item[$(ii)$]  if $\forall a\in V\setminus S,~|N_s(a)\cap S|> 2,$ then $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyperTree-Decomposition;
   \item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is an 2-SuperHyperDefensive  SuperHyperTree-Decomposition;
     \item[$(iv)$]  if $\forall a\in V\setminus S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive  SuperHyperTree-Decomposition.
\end{itemize}
\end{proposition}

            
 \section{Extreme Applications in Cancer's Extreme Recognition}
The cancer is the Extreme disease but the Extreme model is going to figure out what's going on this Extreme phenomenon. The special Extreme case of this Extreme 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 Extreme recognition of the cancer could help to find some Extreme treatments for this Extreme disease.
\\
In the following, some Extreme steps are Extreme devised on this disease.
\begin{description}
 \item[Step 1. (Extreme Definition)] The Extreme recognition of the cancer in the long-term Extreme function.
  \item[Step 2. (Extreme Issue)] The specific region has been assigned by the Extreme model [it's called Extreme SuperHyperGraph] and the long Extreme 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 Extreme SuperHyperGraph] to have convenient perception on what's happened and what's done.
 \item[Step 3. (Extreme Model)]  
There are some specific Extreme models, which are well-known and they've got the names, and some general Extreme models. The moves and the Extreme traces of the cancer on the complex tracks and between complicated groups of cells could be fantasized by a Extreme SuperHyperPath(-/SuperHyperTree-Decomposition, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite,  SuperHyperWheel). The aim is to find either the Extreme   SuperHyperTree-Decomposition or the Extreme   SuperHyperTree-Decomposition in those Extreme Extreme SuperHyperModels.
  \section{Case 1: The Initial Extreme Steps Toward Extreme SuperHyperBipartite as  Extreme SuperHyperModel}

 \item[Step 4. (Extreme Solution)]
In the Extreme Figure \eqref{136NSHGaa21aa}, the Extreme SuperHyperBipartite is Extreme highlighted and Extreme featured.
\begin{figure}
 \includegraphics[width=100mm]{136NSHG21.png}
 \caption{a Extreme  SuperHyperBipartite Associated to the Notions of   Extreme SuperHyperTree-Decomposition}
 \label{136NSHGaa21aa}
\end{figure}
\\ By using the Extreme Figure \eqref{136NSHGaa21aa} and the Table \eqref{136TBLaa21aa}, the Extreme SuperHyperBipartite is obtained.
\\
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{136NSHGaa21aa},
is the Extreme SuperHyperTree-Decomposition.
\begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Extreme SuperHyperBipartite}
\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{136TBLaa21aa}
\end{table}
    \section{Case 2: The Increasing Extreme Steps Toward Extreme SuperHyperMultipartite as Extreme SuperHyperModel}

     \item[Step 4. (Extreme Solution)]
In the Extreme Figure \eqref{136NSHGaa22aa}, the Extreme SuperHyperMultipartite is Extreme highlighted and  Extreme featured.
\begin{figure}
 \includegraphics[width=100mm]{136NSHG22.png}
 \caption{a Extreme  SuperHyperMultipartite Associated to the Notions of   Extreme SuperHyperTree-Decomposition}
 \label{136NSHGaa22aa}
\end{figure}
\\
 By using the Extreme Figure \eqref{136NSHGaa22aa} and the Table \eqref{136TBLaa22aa}, the Extreme SuperHyperMultipartite is obtained.
 \\
 The obtained Extreme SuperHyperSet, by the Extreme Algorithm in previous result, of the Extreme SuperHyperVertices of the connected Extreme SuperHyperMultipartite $ESHM:(V,E),$ in the Extreme SuperHyperModel \eqref{136NSHGaa22aa}, is the  Extreme SuperHyperTree-Decomposition.
\begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Extreme SuperHyperMultipartite}
\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{136TBLaa22aa}
\end{table}
     \end{description}
\section{Wondering Open Problems But As The Directions To Forming The Motivations}
In what follows, some ``problems'' and some ``questions'' are proposed.
\\
The   SuperHyperTree-Decomposition and the Extreme   SuperHyperTree-Decomposition are defined on a real-world application, titled ``Cancer's Recognitions''.
\begin{question}
Which the else SuperHyperModels could be defined based on Cancer's recognitions?
\end{question}
\begin{question}
Are there some SuperHyperNotions related to   SuperHyperTree-Decomposition and the Extreme   SuperHyperTree-Decomposition?
\end{question}
\begin{question}
Are there some Algorithms to be defined on the SuperHyperModels to compute them?
\end{question}
\begin{question}
Which the SuperHyperNotions are related to beyond the   SuperHyperTree-Decomposition and the Extreme   SuperHyperTree-Decomposition?
\end{question}
\begin{problem}
The   SuperHyperTree-Decomposition and the Extreme   SuperHyperTree-Decomposition do a SuperHyperModel for the Cancer's recognitions and they're based on   SuperHyperTree-Decomposition, are there else?
\end{problem}
\begin{problem}
Which the fundamental SuperHyperNumbers are related to these SuperHyperNumbers types-results?
\end{problem}
\begin{problem}
What's the independent research based on Cancer's recognitions concerning the multiple types of SuperHyperNotions?
\end{problem}
\section{Conclusion and  Closing Remarks}
In this section, concluding remarks and closing remarks are represented. The drawbacks of this research are illustrated. Some benefits and some advantages of this research are highlighted.
\\
This research uses some approaches to make Extreme SuperHyperGraphs more understandable. In this endeavor, two SuperHyperNotions are defined on the   SuperHyperTree-Decomposition. For that sake in the second definition, the main definition of the Extreme SuperHyperGraph is redefined on the position of the alphabets. Based on the new definition for the Extreme SuperHyperGraph, the new SuperHyperNotion, Extreme   SuperHyperTree-Decomposition, finds the convenient background to implement some results based on that. Some SuperHyperClasses and some Extreme SuperHyperClasses are the cases of this research on the modeling of the regions where are under the attacks of the cancer to recognize this disease as it's mentioned on the title ``Cancer's Recognitions''. To formalize the instances on the SuperHyperNotion,   SuperHyperTree-Decomposition, the new SuperHyperClasses and SuperHyperClasses, are introduced. Some general results are gathered in the section on the   SuperHyperTree-Decomposition and the Extreme   SuperHyperTree-Decomposition. The clarifications, instances and literature reviews have taken the whole way through. In this research, the literature reviews have fulfilled the lines containing the notions and the results. The SuperHyperGraph and Extreme SuperHyperGraph are the SuperHyperModels on the ``Cancer's Recognitions'' 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 longest and strongest styles with the formation of the design and the architecture are formally called ``  SuperHyperTree-Decomposition'' 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.
\begin{table}[ht]
\centering
\caption{An Overlook On This Research And Beyond}
\label{136TBLTBL}
\begin{tabular}[t]{|c|c|}
\hline
\textcolor{black}{Advantages}&\textcolor{black}{Limitations}\\
\hline
\textcolor{black}{1. }\textcolor{red}{Redefining  Extreme SuperHyperGraph} &\textcolor{black}{1. }\textcolor{blue}{General Results} \\  &
\\
 \textcolor{black}{2. }\textcolor{red}{  SuperHyperTree-Decomposition}& \\ &
\\
\textcolor{black}{3. } \textcolor{red}{Extreme   SuperHyperTree-Decomposition}  &\textcolor{black}{2. } \textcolor{blue}{Other SuperHyperNumbers}
\\&
\\
\textcolor{black}{4. }\textcolor{red}{Modeling of Cancer's Recognitions}    &  \\&
\\
\textcolor{black}{5. }\textcolor{red}{SuperHyperClasses}    &\textcolor{black}{3. }\textcolor{blue}{SuperHyperFamilies}   \\
\hline
\end{tabular}
\end{table}
In the Table \eqref{136TBLTBL}, benefits and avenues for this research are, figured out, pointed out and spoken out.
      
      \section{
Extreme SuperHyperDuality But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Different Extreme Types of Extreme SuperHyperDuality).\\
 Assume a Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Extreme SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called
\begin{itemize}
  \item[$(i)$] \textbf{Extreme e-SuperHyperDuality} if $\forall E_i\in E',~\exists E_j\in E_{ESHG:(V,E)}\setminus E'$ such that $V_a\in E_i,E_j;$
\item[$(ii)$] \textbf{Extreme re-SuperHyperDuality} if $\forall E_i\in E',~\exists E_j\in E_{ESHG:(V,E)}\setminus E'$ such that $V_a\in E_i,E_j$ and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$
 \item[$(iii)$] \textbf{Extreme v-SuperHyperDuality} if $\forall V_i\in V',~\exists V_j\in V_{ESHG:(V,E)}\setminus V'$ such that $V_i,V_j\in E_a;$
\item[$(iv)$]  \textbf{Extreme rv-SuperHyperDuality} if $\forall V_i\in V',~\exists V_j\in V_{ESHG:(V,E)}\setminus V'$ such that $V_i,V_j\in E_a$ and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(v)$] \textbf{Extreme SuperHyperDuality} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality.
\end{itemize}
\end{definition}

\begin{definition}((Extreme) SuperHyperDuality).\\
 Assume a Extreme SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Extreme SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
 \item[$(i)$]
an \textbf{Extreme SuperHyperDuality} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality;
 \item[$(ii)$]
a \textbf{Extreme SuperHyperDuality} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for a  Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of the Extreme SuperHyperEdges of a Extreme SuperHyperSet $S$ of high Extreme cardinality  consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality;
 \item[$(iii)$]
an \textbf{Extreme SuperHyperDuality SuperHyperPolynomial} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(iv)$]
a \textbf{Extreme SuperHyperDuality SuperHyperPolynomial} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality; and the Extreme power is corresponded to its Extreme coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperDuality} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality;
 \item[$(vi)$]
a \textbf{Extreme R-SuperHyperDuality} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of the Extreme SuperHyperVertices of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality;
 \item[$(vii)$]
an \textbf{Extreme R-SuperHyperDuality SuperHyperPolynomial} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(viii)$]
a \textbf{Extreme SuperHyperDuality SuperHyperPolynomial} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is  the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality; and the Extreme power is corresponded to its Extreme coefficient.
\end{itemize}
\end{definition}
\begin{example}\label{136EXM1}
 Assume a Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Extreme Figures in every Extreme items.
 \begin{itemize}
 \item On the Figure \eqref{136NSHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.  $E_1$ and $E_3$ are some empty Extreme  
 SuperHyperEdges but $E_2$ is a loop Extreme SuperHyperEdge and $E_4$ is a 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 a Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Extreme SuperHyperDuality.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.  $E_1,E_2$ and $E_3$ are some empty Extreme SuperHyperEdges but $E_4$ is a 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 a Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Extreme SuperHyperDuality.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG3}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG4}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_4,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=2z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG5}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=4z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG6}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_{{3i+1}_{i=0}^3},E_{{3i+24}_{i=0}^3}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}6z^8.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_{{3i+1}_{i=0}^7}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=6z^8.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG7},  the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_{15},E_{16},E_{17}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_3,V_{13},V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
 \\&&
4\times5\times5 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG8}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_3,V_{13},V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
 \\&&
4\times5\times5 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG9}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_{{3i+1}_{i=0}^3},E_{23}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=3z^5.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_{{3i+1}_{i=0}^3},V_{15}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=3z^5.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG10}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_3,V_{13},V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
  \\&&
4\times5\times5z^3.
\end{eqnarray*}

   \item On the Figure \eqref{136NSHG11}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_1,E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_6,V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
  \\&&
3\times3z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG12}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_1,V^{i\neq 5,7,8}_{i_{i=4}^{10}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
5z^5.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG13}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_5,E_9\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_1,V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
 \\&&
 3\times3 z^2.
\end{eqnarray*}

   \item On the Figure \eqref{136NSHG14}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=2z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
z.
\end{eqnarray*}

   \item On the Figure \eqref{136NSHG15}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
z.
\end{eqnarray*}

   \item On the Figure \eqref{136NSHG16}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
 \\&&
 (2\times1\times2)+(2\times4\times5)z.
\end{eqnarray*}

   \item On the Figure \eqref{136NSHG17}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
 \\&&
 (1\times1\times2)z.
\end{eqnarray*}

   \item On the Figure \eqref{136NSHG18}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
 \\&&
 (2\times2\times2)z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG19}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_{{3i+1}_{i=0^3}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=3z^4.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_{{2i+1}_{i=0^5}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
2z^6.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG20}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=10z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=z.
\end{eqnarray*}

  \item On the Figure \eqref{95NHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=2z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=10z.
\end{eqnarray*}

  \item On the Figure \eqref{95NHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=4z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_3,V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}
  \\&&
=10\times9+10\times6+12\times9+12\times6z^2.
\end{eqnarray*}

\end{itemize}
\end{example}
The previous Extreme approach apply on the upcoming Extreme results on Extreme SuperHyperClasses.
\begin{proposition}
Assume a connected Extreme SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=
         \\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}
 \\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \end{eqnarray*}
be a longest path taken from a connected Extreme SuperHyperPath $ESHP:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperDuality. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM18a}
In the Figure \eqref{136NSHG18a}, the connected Extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Extreme SuperHyperSet,  in the Extreme SuperHyperModel \eqref{136NSHG18a}, is the  SuperHyperDuality.

\end{example}

\begin{proposition}
Assume a connected Extreme SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=
         \\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}
 \\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \end{eqnarray*}
be a longest path taken from a connected Extreme SuperHyperCycle $ESHC:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperDuality. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM19a}
In the Figure \eqref{136NSHG19a}, the connected Extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Extreme SuperHyperSet, in the Extreme SuperHyperModel \eqref{136NSHG19a},
is the Extreme  SuperHyperDuality.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperStar $ESHS:(V,E).$ Then  
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperDuality}}=\{E\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperDuality   SuperHyperPolynomial}}
\\&&=|i~|~E_i\in E_{ESHG:(V,E)|_{\text{Extreme Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperDuality}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperDuality SuperHyperPolynomial}}=z.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
  \begin{eqnarray*}
 && P:V^{EXTERNAL}_i,E_i,CENTER,V^{EXTERNAL}_j.
 \end{eqnarray*}
  be a longest path taken a connected Extreme SuperHyperStar $ESHS:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperDuality. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM20a}
In the Figure \eqref{136NSHG20a}, 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{136NSHG20a}, is the  Extreme SuperHyperDuality.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperDuality}}
\\&&=\{E_i\in E_{{P_i}^{ESHG:(V,E)}},~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperDuality SuperHyperPolynomial}}
\\&&=(\sum_{i=|{P}^{ESHG:(V,E)}|} (\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|)\text{choose} |{P_i}^{ESHG:(V,E)}|)
\\&&
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperDuality}}
\\&&=\{V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperDuality SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}

\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperBipartite $ESHB:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperDuality. The latter is straightforward. Then there's at least one SuperHyperDuality. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperDuality could be applied. There are only two SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest SuperHyperDuality taken from a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-of-SuperHyperPart SuperHyperEdges are attained in any solution
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM21a}
In the Extreme Figure \eqref{136NSHG21a}, 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{136NSHG21a}, is the Extreme  SuperHyperDuality.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperDuality}}
\\&&=\{E_i\in E_{{P_i}^{ESHG:(V,E)}},~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperDuality SuperHyperPolynomial}}
\\&&=(\sum_{i=|{P}^{ESHG:(V,E)}|} (\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|)\text{choose} |{P_i}^{ESHG:(V,E)}|)
\\&&
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperDuality}}
\\&&=\{V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperDuality SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
is a longest SuperHyperDuality taken from a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperDuality. The latter is straightforward.  Then there's at least one SuperHyperDuality. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperDuality could be applied. There are only $z'$ SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the  
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-of-SuperHyperPart SuperHyperEdges are attained in any solution
    \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, 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{136NSHG22a}, is the  Extreme SuperHyperDuality.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperDuality}}=\{E^{*}\in E^{*}_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperDuality   SuperHyperPolynomial}}
\\&&=|i~|~E^{*}_i\in E^{*}_{ESHG:(V,E)|_{\text{Extreme Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperDuality}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperDuality SuperHyperPolynomial}}=z.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E^{*}_1,
 \\&&V^{EXTERNAL}_2,E^{*}_2,
 \\&&\ldots,
 \\&&E^{*}_{|E^{*}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}},V^{EXTERNAL}_{|E^{*}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}+1}
 \end{eqnarray*}
 is a longest SuperHyperDuality taken from a connected Extreme SuperHyperWheel $ESHW:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E^{*}_z\in E^{*}_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E^{*}_z
\equiv
\\&& \exists! E^{*}_z\in E^{*}_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E^{*}_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperDuality. The latter is straightforward. Then there's at least one SuperHyperDuality. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperDuality could be applied. The unique embedded SuperHyperDuality proposes some longest SuperHyperDuality excerpt from some representatives. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM23a}
In the Extreme Figure \eqref{136NSHG23a}, 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{136NSHG23a}, is the  Extreme SuperHyperDuality.
            \end{example}
            \section{
Extreme SuperHyperJoin But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Different Extreme Types of Extreme SuperHyperJoin).\\
 Assume a Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Extreme SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called
\begin{itemize}
  \item[$(i)$] \textbf{Extreme e-SuperHyperJoin} if $\forall E_i\in E_{ESHG:(V,E)}\setminus E',~\exists E_j\in E',$ such that $V_a\in E_i,E_j;$ and $\forall E_i,E_j\in E',$ such that $V_a\not\in E_i,E_j;$
\item[$(ii)$] \textbf{Extreme re-SuperHyperJoin} if $\forall E_i\in E_{ESHG:(V,E)}\setminus E',~\exists E_j\in E',$ such that $V_a\in E_i,E_j;$ $\forall E_i,E_j\in E',$ such that $V_a\not\in E_i,E_j;$ and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$
 \item[$(iii)$] \textbf{Extreme v-SuperHyperJoin} if $\forall V_i\in E_{ESHG:(V,E)}\setminus V',~\exists V_j\in V',$ such that $V_i,V_j\not\in E_a;$ and $\forall V_i,V_j\in V',$ such that $ V_i,V_j\not\in E_a;$
\item[$(iv)$]  \textbf{Extreme rv-SuperHyperJoin} if $\forall V_i\in E_{ESHG:(V,E)}\setminus V',~\exists V_j\in V',$ such that $V_i,V_j\in E_a;$ $\forall V_i,V_j\in V',$ such that $ V_i,V_j\not\in E_a;$ and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(v)$] \textbf{Extreme SuperHyperJoin} if it's either of Extreme e-SuperHyperJoin, Extreme re-SuperHyperJoin, Extreme v-SuperHyperJoin, and Extreme rv-SuperHyperJoin.
\end{itemize}
\end{definition}

\begin{definition}((Extreme) SuperHyperJoin).\\
 Assume a Extreme SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Extreme SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
 \item[$(i)$]
an \textbf{Extreme SuperHyperJoin} if it's either of Extreme e-SuperHyperJoin, Extreme re-SuperHyperJoin, Extreme v-SuperHyperJoin, and Extreme rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin;
 \item[$(ii)$]
a \textbf{Extreme SuperHyperJoin} if it's either of Extreme e-SuperHyperJoin, Extreme re-SuperHyperJoin, Extreme v-SuperHyperJoin, and Extreme rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for a  Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of the Extreme SuperHyperEdges of a Extreme SuperHyperSet $S$ of high Extreme cardinality  consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin;
 \item[$(iii)$]
an \textbf{Extreme SuperHyperJoin SuperHyperPolynomial} if it's either of Extreme e-SuperHyperJoin, Extreme re-SuperHyperJoin, Extreme v-SuperHyperJoin, and Extreme rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(iv)$]
a \textbf{Extreme SuperHyperJoin SuperHyperPolynomial} if it's either of Extreme e-SuperHyperJoin, Extreme re-SuperHyperJoin, Extreme v-SuperHyperJoin, and Extreme rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin; and the Extreme power is corresponded to its Extreme coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperJoin} if it's either of Extreme e-SuperHyperJoin, Extreme re-SuperHyperJoin, Extreme v-SuperHyperJoin, and Extreme rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin;
 \item[$(vi)$]
a \textbf{Extreme R-SuperHyperJoin} if it's either of Extreme e-SuperHyperJoin, Extreme re-SuperHyperJoin, Extreme v-SuperHyperJoin, and Extreme rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of the Extreme SuperHyperVertices of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin;
 \item[$(vii)$]
an \textbf{Extreme R-SuperHyperJoin SuperHyperPolynomial} if it's either of Extreme e-SuperHyperJoin, Extreme re-SuperHyperJoin, Extreme v-SuperHyperJoin, and Extreme rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(viii)$]
a \textbf{Extreme SuperHyperJoin SuperHyperPolynomial} if it's either of Extreme e-SuperHyperJoin, Extreme re-SuperHyperJoin, Extreme v-SuperHyperJoin, and Extreme rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is  the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin; and the Extreme power is corresponded to its Extreme coefficient.
\end{itemize}
\end{definition}

\begin{example}\label{136EXM1}
 Assume a Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Extreme Figures in every Extreme items.
 \begin{itemize}
 \item On the Figure \eqref{136NSHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.  $E_1$ and $E_3$ are some empty Extreme  
 SuperHyperEdges but $E_2$ is a loop Extreme SuperHyperEdge and $E_4$ is a 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 a Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Extreme SuperHyperJoin.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.  $E_1,E_2$ and $E_3$ are some empty Extreme SuperHyperEdges but $E_4$ is a 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 a Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Extreme SuperHyperJoin.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG3}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG4}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_4,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=2z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG5}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=4z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG6}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_{{3i+1}_{i=0}^3},E_{{3i+24}_{i=0}^3}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}6z^8.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_{{3i+1}_{i=0}^7}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=6z^8.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG7},  the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_{15},E_{16},E_{17}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_3,V_{13},V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
 \\&&
4\times5\times5 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG8}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_3,V_{13},V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
 \\&&
4\times5\times5 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG9}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_{{3i+1}_{i=0}^3},E_{23}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^5.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_{{3i+1}_{i=0}^3},V_{15}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=3z^5.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG10}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_3,V_{13},V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
  \\&&
4\times5\times5z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG11}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_1,E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_6,V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
  \\&&
3\times3z^2.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG12}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_1,V^{i\neq 5,7,8}_{i_{i=4}^{10}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
5z^5.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG13}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_3,E_9\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_1,V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
 \\&&
 3\times3 z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG14}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=2z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG15}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG16}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_2,V_7,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
 \\&&
 (1\times5\times5)+(1\times2+1)z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG17}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_{27},V_2,V_7,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
 \\&&
 (1\times1\times2+1)z^4.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG18}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_{27},V_2,V_7,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
 \\&&
 (1\times1\times2+1)z^4.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG19}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_{{3i+1}_{i=0^3}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^4.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_{{2i+1}_{i=0^5}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
2z^6.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG20}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=10z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=z.
\end{eqnarray*}

 \item On the Figure \eqref{95NHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=2z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=10z.
\end{eqnarray*}
 \item On the Figure \eqref{95NHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperJoin, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=\{E_2,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}=\{V_3,V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}
  \\&&
=10\times6+10\times6+12\times6+12\times6z^2.
\end{eqnarray*}
\end{itemize}
\end{example}
The previous Extreme approach apply on the upcoming Extreme results on Extreme SuperHyperClasses.
\begin{proposition}
Assume a connected Extreme SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=
         \\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}
 \\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \end{eqnarray*}
be a longest path taken from a connected Extreme SuperHyperPath $ESHP:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperJoin. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM18a}
In the Figure \eqref{136NSHG18a}, the connected Extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Extreme SuperHyperSet,  in the Extreme SuperHyperModel \eqref{136NSHG18a}, is the  SuperHyperJoin.
\end{example}

\begin{proposition}
Assume a connected Extreme SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=
         \\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}
 \\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \end{eqnarray*}
be a longest path taken from a connected Extreme SuperHyperCycle $ESHC:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperJoin. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM19a}
In the Figure \eqref{136NSHG19a}, the connected Extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Extreme SuperHyperSet, in the Extreme SuperHyperModel \eqref{136NSHG19a},
is the Extreme  SuperHyperJoin.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperStar $ESHS:(V,E).$ Then  
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperJoin}}=\{E\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperJoin   SuperHyperPolynomial}}
\\&&=|i~|~E_i\in E_{ESHG:(V,E)|_{\text{Extreme Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperJoin}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperJoin SuperHyperPolynomial}}=z.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
  \begin{eqnarray*}
 && P:V^{EXTERNAL}_i,E_i,CENTER,V^{EXTERNAL}_j.
 \end{eqnarray*}
  be a longest path taken a connected Extreme SuperHyperStar $ESHS:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperJoin. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM20a}
In the Figure \eqref{136NSHG20a}, 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{136NSHG20a}, is the  Extreme SuperHyperJoin.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperJoin}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&
\{E_i\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperJoin}}
\\&&
=\text{(OTHERWISE)}.
\\&&
\{\},
\\&& \text{If}
~\exists {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|\neq\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&=(\sum_{i=|{P}^{ESHG:(V,E)}|} (\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|)\text{choose} |{P_i}^{ESHG:(V,E)}|)
\\&&
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}
\\&&
=\text{(OTHERWISE)}0.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperJoin}}
\\&&=\{V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperJoin SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}

\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperBipartite $ESHB:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperJoin. The latter is straightforward. Then there's no at least one SuperHyperJoin. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperJoin could be applied. There are only two SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest SuperHyperJoin taken from a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-of-SuperHyperPart SuperHyperEdges are attained in any solution
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM21a}
In the Extreme Figure \eqref{136NSHG21a}, 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{136NSHG21a}, is the Extreme  SuperHyperJoin.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperJoin}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&
\{E_i\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperJoin}}
\\&&
=\text{(OTHERWISE)}.
\\&&
\{\},
\\&& \text{If}
~\exists {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|\neq\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&=(\sum_{i=|{P}^{ESHG:(V,E)}|} (\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|)\text{choose} |{P_i}^{ESHG:(V,E)}|)
\\&&
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}
\\&&
=\text{(OTHERWISE)}0.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperJoin}}
\\&&=\{V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperJoin SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
is a longest SuperHyperJoin taken from a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperJoin. The latter is straightforward.  Then there's no at least one SuperHyperJoin. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperJoin could be applied. There are only $z'$ SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the  
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-of-SuperHyperPart SuperHyperEdges are attained in any solution
    \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, 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{136NSHG22a}, is the  Extreme SuperHyperJoin.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin}}=
         \\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperJoin SuperHyperPolynomial}}
 \\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperJoin}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperJoin SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \end{eqnarray*}
  is a longest SuperHyperJoin taken from a connected Extreme SuperHyperWheel $ESHW:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperJoin. The latter is straightforward. Then there's at least one SuperHyperJoin. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperJoin could be applied. The unique embedded SuperHyperJoin proposes some longest SuperHyperJoin excerpt from some representatives. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM23a}
In the Extreme Figure \eqref{136NSHG23a}, 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{136NSHG23a}, is the  Extreme SuperHyperJoin.

            \end{example}
           \section{
Extreme SuperHyperPerfect But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Different Extreme Types of Extreme SuperHyperPerfect).\\
 Assume a Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Extreme SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called
\begin{itemize}
  \item[$(i)$] \textbf{Extreme e-SuperHyperPerfect} if $\forall E_i\in E_{ESHG:(V,E)}\setminus E',~\exists! E_j\in E',$ such that $V_a\in E_i,E_j;$
\item[$(ii)$] \textbf{Extreme re-SuperHyperPerfect} if $\forall E_i\in E_{ESHG:(V,E)}\setminus E',~\exists! E_j\in E',$ such that $V_a\in E_i,E_j;$  and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$
 \item[$(iii)$] \textbf{Extreme v-SuperHyperPerfect} if $\forall V_i\in V_{ESHG:(V,E)}\setminus V',~\exists! V_j\in V',$ such that $V_i,V_j\in E_a;$
\item[$(iv)$]  \textbf{Extreme rv-SuperHyperPerfect} if $\forall V_i\in V_{ESHG:(V,E)}\setminus V',~\exists! V_j\in V',$ such that $V_i,V_j\in E_a;$ and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(v)$] \textbf{Extreme SuperHyperPerfect} if it's either of Extreme e-SuperHyperPerfect, Extreme re-SuperHyperPerfect, Extreme v-SuperHyperPerfect, and Extreme rv-SuperHyperPerfect.
\end{itemize}
\end{definition}

\begin{definition}((Extreme) SuperHyperPerfect).\\
 Assume a Extreme SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Extreme SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
 \item[$(i)$]
an \textbf{Extreme SuperHyperPerfect} if it's either of Extreme e-SuperHyperPerfect, Extreme re-SuperHyperPerfect, Extreme v-SuperHyperPerfect, and Extreme rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperPerfect;
 \item[$(ii)$]
a \textbf{Extreme SuperHyperPerfect} if it's either of Extreme e-SuperHyperPerfect, Extreme re-SuperHyperPerfect, Extreme v-SuperHyperPerfect, and Extreme rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for a  Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of the Extreme SuperHyperEdges of a Extreme SuperHyperSet $S$ of high Extreme cardinality  consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperPerfect;
 \item[$(iii)$]
an \textbf{Extreme SuperHyperPerfect SuperHyperPolynomial} if it's either of Extreme e-SuperHyperPerfect, Extreme re-SuperHyperPerfect, Extreme v-SuperHyperPerfect, and Extreme rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperPerfect; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(iv)$]
a \textbf{Extreme SuperHyperPerfect SuperHyperPolynomial} if it's either of Extreme e-SuperHyperPerfect, Extreme re-SuperHyperPerfect, Extreme v-SuperHyperPerfect, and Extreme rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperPerfect; and the Extreme power is corresponded to its Extreme coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperPerfect} if it's either of Extreme e-SuperHyperPerfect, Extreme re-SuperHyperPerfect, Extreme v-SuperHyperPerfect, and Extreme rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperPerfect;
 \item[$(vi)$]
a \textbf{Extreme R-SuperHyperPerfect} if it's either of Extreme e-SuperHyperPerfect, Extreme re-SuperHyperPerfect, Extreme v-SuperHyperPerfect, and Extreme rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of the Extreme SuperHyperVertices of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperPerfect;
 \item[$(vii)$]
an \textbf{Extreme R-SuperHyperPerfect SuperHyperPolynomial} if it's either of Extreme e-SuperHyperPerfect, Extreme re-SuperHyperPerfect, Extreme v-SuperHyperPerfect, and Extreme rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperPerfect; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(viii)$]
a \textbf{Extreme SuperHyperPerfect SuperHyperPolynomial} if it's either of Extreme e-SuperHyperPerfect, Extreme re-SuperHyperPerfect, Extreme v-SuperHyperPerfect, and Extreme rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is  the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperPerfect; and the Extreme power is corresponded to its Extreme coefficient.
\end{itemize}
\end{definition}

\begin{example}\label{136EXM1}
 Assume a Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Extreme Figures in every Extreme items.
 \begin{itemize}
 \item On the Figure \eqref{136NSHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.  $E_1$ and $E_3$ are some empty Extreme  
 SuperHyperEdges but $E_2$ is a loop Extreme SuperHyperEdge and $E_4$ is a 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 a Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Extreme SuperHyperPerfect.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.  $E_1,E_2$ and $E_3$ are some empty Extreme SuperHyperEdges but $E_4$ is a 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 a Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Extreme SuperHyperPerfect.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG3}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
 \item On the Figure \eqref{136NSHG4}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_4,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=2z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG5}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=4z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG6}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_{{3i+1}_{i=0}^3},E_{{3i+24}_{i=0}^3}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}6z^8.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_{{3i+1}_{i=0}^7}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=6z^8.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG7},  the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_{15},E_{16},E_{17}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_3,V_6,V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
 \\&&
3\times4\times4 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG8}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_3,V_6,V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
 \\&&
3\times4\times4 z^3.
\end{eqnarray*}
 \item On the Figure \eqref{136NSHG9}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperPerfect}}=\{E_{{3i+1}_{i=0}^3},E_{23}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperPerfect SuperHyperPolynomial}}=3z^5.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperPerfect}}=\{V_{{3i+1}_{i=0}^3},V_{15}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=3z^5.
\end{eqnarray*}
 \item On the Figure \eqref{136NSHG10}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_3,V_6,V_8\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
  \\&&
3\times4\times4z^3.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG11}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_1,E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_6,V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
  \\&&
3\times2z^2.
\end{eqnarray*}
 \item On the Figure \eqref{136NSHG12}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_1,V^{i\neq 5,7,8}_{i_{i=4}^{10}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
5z^5.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG13}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_3,E_9\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_1,V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
 \\&&
 3\times3 z^2.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG14}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_1\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=2z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG15}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
z.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG16}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperPerfect}}=\{V_2,V_7,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=
 \\&&
 (1\times5\times5)+(1\times2+1)z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG17}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperPerfect}}=\{V_{27},V_2,V_7,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=
 \\&&
 (1\times1\times2+1)z^4.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG18}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=3z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperPerfect}}=\{V_{27},V_2,V_7,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=
 \\&&
 (1\times1\times2+1)z^4.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG19}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_{{3i+1}_{i=0^3}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=3z^4.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperPerfect}}=\{V_{{2i+1}_{i=0^5}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=
2z^6.
\end{eqnarray*}
  \item On the Figure \eqref{136NSHG20}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=10z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=z.
\end{eqnarray*}
 \item On the Figure \eqref{95NHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=2z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=10z.
\end{eqnarray*}
 \item On the Figure \eqref{95NHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperPerfect, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E_2,E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperPerfect}}=\{V_3,V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}
  \\&&
=10\times6+10\times6+12\times6+12\times6z^2.
\end{eqnarray*}
\end{itemize}
\end{example}
The previous Extreme approach apply on the upcoming Extreme results on Extreme SuperHyperClasses.
\begin{proposition}
Assume a connected Extreme SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=
         \\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}
 \\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \end{eqnarray*}
be a longest path taken from a connected Extreme SuperHyperPath $ESHP:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperPerfect. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM18a}
In the Figure \eqref{136NSHG18a}, the connected Extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Extreme SuperHyperSet,  in the Extreme SuperHyperModel \eqref{136NSHG18a}, is the  SuperHyperPerfect.
\end{example}

\begin{proposition}
Assume a connected Extreme SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=
         \\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}
 \\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}{3}}.
 \end{eqnarray*}
be a longest path taken from a connected Extreme SuperHyperCycle $ESHC:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperPerfect. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM19a}
In the Figure \eqref{136NSHG19a}, the connected Extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Extreme SuperHyperSet, in the Extreme SuperHyperModel \eqref{136NSHG19a},
is the Extreme  SuperHyperPerfect.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperStar $ESHS:(V,E).$ Then  
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect   SuperHyperPolynomial}}
\\&&=|i~|~E_i\in E_{ESHG:(V,E)|_{\text{Extreme Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}=z.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
  \begin{eqnarray*}
 && P:V^{EXTERNAL}_i,E_i,CENTER,V^{EXTERNAL}_j.
 \end{eqnarray*}
  be a longest path taken a connected Extreme SuperHyperStar $ESHS:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperPerfect. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM20a}
In the Figure \eqref{136NSHG20a}, 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{136NSHG20a}, is the  Extreme SuperHyperPerfect.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperPerfect}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&
\{E_i\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperPerfect}}
\\&&
=\text{(OTHERWISE)}.
\\&&
\{\},
\\&& \text{If}
~\exists {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|\neq\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&=(\sum_{i=|{P}^{ESHG:(V,E)}|} (\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|)\text{choose} |{P_i}^{ESHG:(V,E)}|)
\\&&
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}
\\&&
=\text{(OTHERWISE)}0.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperPerfect}}
\\&&=\{V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperPerfect SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}

\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperBipartite $ESHB:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperPerfect. The latter is straightforward. Then there's no at least one SuperHyperPerfect. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperPerfect could be applied. There are only two SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest SuperHyperPerfect taken from a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-of-SuperHyperPart SuperHyperEdges are attained in any solution
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM21a}
In the Extreme Figure \eqref{136NSHG21a}, 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{136NSHG21a}, is the Extreme  SuperHyperPerfect.

\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperPerfect}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&
\{E_i\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperPerfect}}
\\&&
=\text{(OTHERWISE)}.
\\&&
\{\},
\\&& \text{If}
~\exists {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|\neq\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}
\\&&=
\text{(PERFECT MATCHING)}.
\\&&=(\sum_{i=|{P}^{ESHG:(V,E)}|} (\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|)\text{choose} |{P_i}^{ESHG:(V,E)}|)
\\&&
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect SuperHyperPolynomial}}
\\&&
=\text{(OTHERWISE)}0.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperPerfect}}
\\&&=\{V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_i\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperPerfect SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
is a longest SuperHyperPerfect taken from a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperPerfect. The latter is straightforward.  Then there's no at least one SuperHyperPerfect. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperPerfect could be applied. There are only $z'$ SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the  
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-of-SuperHyperPart SuperHyperEdges are attained in any solution
    \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2,
 \\&&\ldots,
 \\&&E_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|},V^{EXTERNAL}_{\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|+1}
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, 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{136NSHG22a}, is the  Extreme SuperHyperPerfect.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect}}=\{E\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperPerfect   SuperHyperPolynomial}}
\\&&=|i~|~E_i\in E_{ESHG:(V,E)|_{\text{Extreme Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperPerfect SuperHyperPolynomial}}=z.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
  \begin{eqnarray*}
 && P:V^{EXTERNAL}_i,E_i,CENTER,V^{EXTERNAL}_j.
 \end{eqnarray*}
  is a longest SuperHyperPerfect taken from a connected Extreme SuperHyperWheel $ESHW:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperPerfect. The latter is straightforward. Then there's at least one SuperHyperPerfect. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperPerfect could be applied. The unique embedded SuperHyperPerfect proposes some longest SuperHyperPerfect excerpt from some representatives. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM23a}
In the Extreme Figure \eqref{136NSHG23a}, 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{136NSHG23a}, is the  Extreme SuperHyperPerfect.
            \end{example}

\section{
Extreme SuperHyperTotal But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Different Extreme Types of Extreme SuperHyperTotal).\\
 Assume a Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Extreme SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called
\begin{itemize}
  \item[$(i)$] \textbf{Extreme e-SuperHyperTotal} if $\forall E_i\in E_{ESHG:(V,E)},~\exists! E_j\in E',$ such that $V_a\in E_i,E_j;$
\item[$(ii)$] \textbf{Extreme re-SuperHyperTotal} if $\forall E_i\in E_{ESHG:(V,E)},~\exists! E_j\in E',$ such that $V_a\in E_i,E_j;$  and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$
 \item[$(iii)$] \textbf{Extreme v-SuperHyperTotal} if $\forall V_i\in V_{ESHG:(V,E)},~\exists! V_j\in V',$ such that $V_i,V_j\in E_a;$
\item[$(iv)$]  \textbf{Extreme rv-SuperHyperTotal} if $\forall V_i\in V_{ESHG:(V,E)},~\exists! V_j\in V',$ such that $V_i,V_j\in E_a;$ and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(v)$] \textbf{Extreme SuperHyperTotal} if it's either of Extreme e-SuperHyperTotal, Extreme re-SuperHyperTotal, Extreme v-SuperHyperTotal, and Extreme rv-SuperHyperTotal.
\end{itemize}
\end{definition}

\begin{definition}((Extreme) SuperHyperTotal).\\
 Assume a Extreme SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Extreme SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
 \item[$(i)$]
an \textbf{Extreme SuperHyperTotal} if it's either of Extreme e-SuperHyperTotal, Extreme re-SuperHyperTotal, Extreme v-SuperHyperTotal, and Extreme rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTotal;
 \item[$(ii)$]
a \textbf{Extreme SuperHyperTotal} if it's either of Extreme e-SuperHyperTotal, Extreme re-SuperHyperTotal, Extreme v-SuperHyperTotal, and Extreme rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for a  Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of the Extreme SuperHyperEdges of a Extreme SuperHyperSet $S$ of high Extreme cardinality  consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTotal;
 \item[$(iii)$]
an \textbf{Extreme SuperHyperTotal SuperHyperPolynomial} if it's either of Extreme e-SuperHyperTotal, Extreme re-SuperHyperTotal, Extreme v-SuperHyperTotal, and Extreme rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTotal; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(iv)$]
a \textbf{Extreme SuperHyperTotal SuperHyperPolynomial} if it's either of Extreme e-SuperHyperTotal, Extreme re-SuperHyperTotal, Extreme v-SuperHyperTotal, and Extreme rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTotal; and the Extreme power is corresponded to its Extreme coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperTotal} if it's either of Extreme e-SuperHyperTotal, Extreme re-SuperHyperTotal, Extreme v-SuperHyperTotal, and Extreme rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTotal;
 \item[$(vi)$]
a \textbf{Extreme R-SuperHyperTotal} if it's either of Extreme e-SuperHyperTotal, Extreme re-SuperHyperTotal, Extreme v-SuperHyperTotal, and Extreme rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of the Extreme SuperHyperVertices of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTotal;
 \item[$(vii)$]
an \textbf{Extreme R-SuperHyperTotal SuperHyperPolynomial} if it's either of Extreme e-SuperHyperTotal, Extreme re-SuperHyperTotal, Extreme v-SuperHyperTotal, and Extreme rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTotal; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(viii)$]
a \textbf{Extreme SuperHyperTotal SuperHyperPolynomial} if it's either of Extreme e-SuperHyperTotal, Extreme re-SuperHyperTotal, Extreme v-SuperHyperTotal, and Extreme rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is  the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTotal; and the Extreme power is corresponded to its Extreme coefficient.
\end{itemize}
\end{definition}

\begin{example}\label{136EXM1}
 Assume a Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Extreme Figures in every Extreme items.
 \begin{itemize}
 \item On the Figure \eqref{136NSHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.  $E_1$ and $E_3$ are some empty Extreme  
 SuperHyperEdges but $E_2$ is a loop Extreme SuperHyperEdge and $E_4$ is a 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 a Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Extreme SuperHyperTotal.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperTotal}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.  $E_1,E_2$ and $E_3$ are some empty Extreme SuperHyperEdges but $E_4$ is a 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 a Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Extreme SuperHyperTotal.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperTotal}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG3}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperTotal}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG4}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-}}=\{E_4,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}=2z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperTotal}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG5}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal}}=\{E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}=4z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperTotal}}=\{V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG6}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_{{i+1}_{i=0}^{9}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}20z^{10}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_{{i+1}_{i=0}^{9}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=20z^{10}.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG7},  the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_{12},E_{13},E_{14}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_{12},V_{13},V_{14}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG8}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_{12},V_{13},V_{14}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
 \\&&
3\times4\times4 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG9}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_{{i+1}_{i=0}^{9}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}10z^{10}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_{{i+1}_{i=0}^{9}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=20z^{10}.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG10}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal}}=\{E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_{12},V_{13},V_{14}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
 \\&&
3\times4\times4 z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG11}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_1,E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=2z^4.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
3z^2.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG12}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=5z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V^{i\neq 4,5,6}_{i_{i=1}^{8}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
z^5.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG13}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_3,E_9,E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=3z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
3z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG14}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V_3\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
2z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG15}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_2,V_3,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG16}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_2,V_6,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
 \\&&
4\times3z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG17}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V_2,V_6,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
 \\&&
4\times3z^4.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG18}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V_2,V_6,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=
 \\&&
2\times4\times3z^4.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG19}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_{{i+2}_{i=0^{11}}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=11z^{10}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperTotal}}=\{V_{{i+2}_{i=0^{11}}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=
11z^{10}.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG20}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_6,E_{10}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=9z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}
   \\&&
  =|(|V|-1)z^2.
\end{eqnarray*}

 \item On the Figure \eqref{95NHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=2z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{V_1,V_2\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=9z^2.
\end{eqnarray*}

 \item On the Figure \eqref{95NHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperTotal, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperTotal}}=\{V_3,V_{10},V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}
  \\&&
=3\times6z^3.
\end{eqnarray*}

\end{itemize}
\end{example}
The previous Extreme approach apply on the upcoming Extreme results on Extreme SuperHyperClasses.
\begin{proposition}
Assume a connected Extreme SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=
         \\&&=\{E_i\}_{i=1}^{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}-2}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}
 \\&&=z^{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}-2}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}-2}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}
z^{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}-2}
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_2,E_2,
 \\&&V^{EXTERNAL}_3,E_3,
 \\&&\ldots
 \end{eqnarray*}
be a longest path taken from a connected Extreme SuperHyperPath $ESHP:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTotal. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM18a}
In the Figure \eqref{136NSHG18a}, the connected Extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Extreme SuperHyperSet,  in the Extreme SuperHyperModel \eqref{136NSHG18a}, is the  SuperHyperTotal.

\end{example}

\begin{proposition}
Assume a connected Extreme SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=
         \\&&=\{E_i\}_{i=1}^{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}-2}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}
 \\&&=({|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}-1)
  \\&&
 z^{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}-2}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}-2}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}
z^{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}-2}
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_2,E_2,
 \\&&V^{EXTERNAL}_3,E_3,
 \\&&\ldots,
 \\&&{E}_{\frac{|E_^{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}-1}},{V^{EXTERNAL}}_{|E_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}-1}}.
 \end{eqnarray*}
 be a longest path taken from a connected Extreme SuperHyperCycle $ESHC:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTotal. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM19a}
In the Figure \eqref{136NSHG19a}, the connected Extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Extreme SuperHyperSet, in the Extreme SuperHyperModel \eqref{136NSHG19a},
is the Extreme  SuperHyperTotal.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperStar $ESHS:(V,E).$ Then  
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_i,E_j\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal   SuperHyperPolynomial}}
\\&&=|i(i-1)~|~E_i\in E_{ESHG:(V,E)|_{\text{Extreme Cardinality}}}|z^2.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{CENTER,V_j\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}=
\\&&
(|V_{ESHG:(V,E)|_{\text{Extreme Cardinality}}}|)~
\text{choose}~
(|V_{ESHG:(V,E)|_{\text{Extreme Cardinality}}}|-1)
\\&&
z^2.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
  \begin{eqnarray*}
 && P:V^{EXTERNAL}_i,E_i,CENTER,E_j.
 \end{eqnarray*}
  be a longest path taken a connected Extreme SuperHyperStar $ESHS:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTotal. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM20a}
In the Figure \eqref{136NSHG20a}, 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{136NSHG20a}, is the  Extreme SuperHyperTotal.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}
\\&&=
\{E_a\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}
\\&&=
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}
\\&&=\{V^{EXTERNAL}_a\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_b\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}

\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperBipartite $ESHB:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTotal. The latter is straightforward. Then there's no at least one SuperHyperTotal. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperTotal could be applied. There are only two SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest SuperHyperTotal taken from a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-of-SuperHyperPart SuperHyperEdges are attained in any solution
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM21a}
In the Extreme Figure \eqref{136NSHG21a}, 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{136NSHG21a}, is the Extreme  SuperHyperTotal.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}
\\&&=
\{E_a\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal SuperHyperPolynomial}}
\\&&=
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}
\\&&=\{V^{EXTERNAL}_a\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_b\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}

\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest SuperHyperTotal taken from a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTotal. The latter is straightforward.  Then there's no at least one SuperHyperTotal. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperTotal could be applied. There are only $z'$ SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the  
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-of-SuperHyperPart SuperHyperEdges are attained in any solution
    \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, 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{136NSHG22a}, is the  Extreme SuperHyperTotal.

\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal}}=\{E_i,E_j\in E^{*}_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperTotal   SuperHyperPolynomial}}
\\&&=|i(i-1)~|~E_i\in E^{*}_{ESHG:(V,E)|_{\text{Extreme Cardinality}}}|z^2.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal}}=\{CENTER,V_j\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperTotal SuperHyperPolynomial}}=
\\&&
(|V_{ESHG:(V,E)|_{\text{Extreme Cardinality}}}|)~
\text{choose}~
(|V_{ESHG:(V,E)|_{\text{Extreme Cardinality}}}|-1)
\\&&
z^2.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:V^{EXTERNAL}_i,E^{*}_i,CENTER,E_j.
 \end{eqnarray*}
  is a longest SuperHyperTotal taken from a connected Extreme SuperHyperWheel $ESHW:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperTotal. The latter is straightforward. Then there's at least one SuperHyperTotal. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperTotal could be applied. The unique embedded SuperHyperTotal proposes some longest SuperHyperTotal excerpt from some representatives. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM23a}
In the Extreme Figure \eqref{136NSHG23a}, 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{136NSHG23a}, is the  Extreme SuperHyperTotal.
            \end{example}
    \section{
Extreme SuperHyperConnected But As The Extensions Excerpt From Dense And Super Forms}        
            
\begin{definition}(Different Extreme Types of Extreme SuperHyperConnected).\\
 Assume a Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Extreme SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called
\begin{itemize}
  \item[$(i)$] \textbf{Extreme e-SuperHyperConnected} if $\forall E_i\in E_{ESHG:(V,E)}\setminus E',~\exists E_j\in E',$ such that $V_a\in E_i,E_j;$ and $\forall E_i,E_j\in E',$ such that $V_a\not\in E_i,E_j;$
\item[$(ii)$] \textbf{Extreme re-SuperHyperConnected} if $\forall E_i\in E_{ESHG:(V,E)}\setminus E',~\exists E_j\in E',$ such that $V_a\in E_i,E_j;$ $\forall E_i,E_j\in E',$ such that $V_a\not\in E_i,E_j;$ and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$
 \item[$(iii)$] \textbf{Extreme v-SuperHyperConnected} if $\forall V_i\in E_{ESHG:(V,E)}\setminus V',~\exists V_j\in V',$ such that $V_i,V_j\not\in E_a;$ and $\forall V_i,V_j\in V',$ such that $ V_i,V_j\not\in E_a;$
\item[$(iv)$]  \textbf{Extreme rv-SuperHyperConnected} if $\forall V_i\in E_{ESHG:(V,E)}\setminus V',~\exists V_j\in V',$ such that $V_i,V_j\in E_a;$ $\forall V_i,V_j\in V',$ such that $ V_i,V_j\not\in E_a;$ and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(v)$] \textbf{Extreme SuperHyperConnected} if it's either of Extreme e-SuperHyperConnected, Extreme re-SuperHyperConnected, Extreme v-SuperHyperConnected, and Extreme rv-SuperHyperConnected.
\end{itemize}
\end{definition}

\begin{definition}((Extreme) SuperHyperConnected).\\
 Assume a Extreme SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a Extreme SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$
Then $E$ is called
\begin{itemize}
 \item[$(i)$]
an \textbf{Extreme SuperHyperConnected} if it's either of Extreme e-SuperHyperConnected, Extreme re-SuperHyperConnected, Extreme v-SuperHyperConnected, and Extreme rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperConnected;
 \item[$(ii)$]
a \textbf{Extreme SuperHyperConnected} if it's either of Extreme e-SuperHyperConnected, Extreme re-SuperHyperConnected, Extreme v-SuperHyperConnected, and Extreme rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for a  Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of the Extreme SuperHyperEdges of a Extreme SuperHyperSet $S$ of high Extreme cardinality  consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperConnected;
 \item[$(iii)$]
an \textbf{Extreme SuperHyperConnected SuperHyperPolynomial} if it's either of Extreme e-SuperHyperConnected, Extreme re-SuperHyperConnected, Extreme v-SuperHyperConnected, and Extreme rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperConnected; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(iv)$]
a \textbf{Extreme SuperHyperConnected SuperHyperPolynomial} if it's either of Extreme e-SuperHyperConnected, Extreme re-SuperHyperConnected, Extreme v-SuperHyperConnected, and Extreme rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperConnected; and the Extreme power is corresponded to its Extreme coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperConnected} if it's either of Extreme e-SuperHyperConnected, Extreme re-SuperHyperConnected, Extreme v-SuperHyperConnected, and Extreme rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperConnected;
 \item[$(vi)$]
a \textbf{Extreme R-SuperHyperConnected} if it's either of Extreme e-SuperHyperConnected, Extreme re-SuperHyperConnected, Extreme v-SuperHyperConnected, and Extreme rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of the Extreme SuperHyperVertices of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperConnected;
 \item[$(vii)$]
an \textbf{Extreme R-SuperHyperConnected SuperHyperPolynomial} if it's either of Extreme e-SuperHyperConnected, Extreme re-SuperHyperConnected, Extreme v-SuperHyperConnected, and Extreme rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperConnected; and the Extreme power is corresponded to its Extreme coefficient;
 \item[$(viii)$]
a \textbf{Extreme SuperHyperConnected SuperHyperPolynomial} if it's either of Extreme e-SuperHyperConnected, Extreme re-SuperHyperConnected, Extreme v-SuperHyperConnected, and Extreme rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is  the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of a Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperConnected; and the Extreme power is corresponded to its Extreme coefficient.
\end{itemize}
\end{definition}

\begin{example}\label{136EXM1}
 Assume a Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Extreme Figures in every Extreme items.
 \begin{itemize}
 \item On the Figure \eqref{136NSHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.  $E_1$ and $E_3$ are some empty Extreme  
 SuperHyperEdges but $E_2$ is a loop Extreme SuperHyperEdge and $E_4$ is a 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 a Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Extreme SuperHyperConnected.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.  $E_1,E_2$ and $E_3$ are some empty Extreme SuperHyperEdges but $E_4$ is a 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 a Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$  \underline{\textbf{is}} excluded in every given Extreme SuperHyperConnected.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG3}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=3z.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG4}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_1,E_2,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_1,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG5}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_3\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=4z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG6}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_{{i+1}_{i=0}^{9}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}20z^{10}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_{{i+1}_{i=0}^{9}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=20z^{10}.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG7},  the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_{12},E_{13},E_{14}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_{12},V_{13},V_{14}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=
 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG8}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_{12},V_{13},V_{14}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=
 \\&&
3\times4\times4 z^3.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG9}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_{{i+1}_{i=0}^{9}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}10z^{10}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_{{i+1}_{i=11}^{19}},V_{22}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=20z^{10}.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG10}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_5\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_{12},V_{13},V_{14}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=
 \\&&
3\times4\times4 z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG11}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_1,E_6,E_7,E_8\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=2z^4.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_1,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=
3z^2.
\end{eqnarray*}

 \item On the Figure \eqref{136NSHG12}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_1,E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=5z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_1,V^{i\neq 4,5,6}_{i_{i=1}^{8}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=
z^5.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG13}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_3,E_9,E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=3z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_1,V_5\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=
3z^2.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG14}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=
z.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG15}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_2,V_3,V_4\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=
z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG16}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_2,V_6,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=
 \\&&
4\times3z^3.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG17}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_1,V_2,V_6,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=
 \\&&
4\times3z^4.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG18}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
&&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_2,E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=z^3.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_1,V_2,V_6,V_{17}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=
 \\&&
2\times4\times3z^4.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG19}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_{{i+2}_{i=0^{11}}}\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=11z^{10}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_{{i+2}_{i=0^{11}}}\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=
11z^{10}.
\end{eqnarray*}

  \item On the Figure \eqref{136NSHG20}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_6\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=10z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}
 =z.
\end{eqnarray*}

 \item On the Figure \eqref{95NHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_2\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}=z.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{V_1\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=10z.
\end{eqnarray*}

 \item On the Figure \eqref{95NHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperConnected, is up. The Extreme Algorithm is Extremely straightforward.
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=\{E_3,E_4\}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=z^2.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}=\{V_3,V_{10},V_6\}.
 \\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}
  \\&&
=3\times6z^3.
\end{eqnarray*}

\end{itemize}
\end{example}
The previous Extreme approach apply on the upcoming Extreme results on Extreme SuperHyperClasses.
\begin{proposition}
Assume a connected Extreme SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=
         \\&&=\{E_i\}_{i=1}^{|{E_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}-2}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}
 \\&&=z^{{|E_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}-2}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{{|E_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}-2}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}
z^{{|E_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}-2}
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_2,E_2,
 \\&&V^{EXTERNAL}_3,E_3,
 \\&&\ldots,
 \\&&{E}_{{|{E}_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}-1},{V^{EXTERNAL}}_{{|{E}_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}-1}.
 \end{eqnarray*}
be a longest path taken from a connected Extreme SuperHyperPath $ESHP:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperConnected. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM18a}
In the Figure \eqref{136NSHG18a}, the connected Extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Extreme SuperHyperSet,  in the Extreme SuperHyperModel \eqref{136NSHG18a}, is the  SuperHyperConnected.
\end{example}

\begin{proposition}
Assume a connected Extreme SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
 &&
          \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}=
         \\&&=\{E_i\}_{i=1}^{{|E_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}-2}.
 \\&&
 \mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}
 \\&&=({|E_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}-1)
  \\&&
 z^{{|E_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}-2}.
 \\&&
   \mathcal{C}(NSHG)_{\text{Extreme R-Quasi-SuperHyperConnected}}
    \\&&=\{V^{EXTERNAL}_i\}_{i=1}^{{|E_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}-2}.
\\&&
  \mathcal{C}(NSHG)_{{\small\text{Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}
  \\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}
z^{{|E_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}-2}
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_2,E_2,
 \\&&V^{EXTERNAL}_3,E_3,
 \\&&\ldots,
 \\&&{E}_{{|E_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}-1},{V^{EXTERNAL}}_{{|E_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}-1}.
 \end{eqnarray*}
 be a longest path taken from a connected Extreme SuperHyperCycle $ESHC:(V,E).$ There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperConnected. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM19a}
In the Figure \eqref{136NSHG19a}, the connected Extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Extreme SuperHyperSet, in the Extreme SuperHyperModel \eqref{136NSHG19a},
is the Extreme  SuperHyperConnected.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperStar $ESHS:(V,E).$ Then  
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_i\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected   SuperHyperPolynomial}}
\\&&=|i~|~E_i\in {|{E}_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperConnected SuperHyperPolynomial}}=
z.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
  \begin{eqnarray*}
 && P:V^{EXTERNAL}_i,E_i,CENTER,E_j.
 \end{eqnarray*}
  be a longest path taken a connected Extreme SuperHyperStar $ESHS:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperConnected. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM20a}
In the Figure \eqref{136NSHG20a}, 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{136NSHG20a}, is the  Extreme SuperHyperConnected.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}
\\&&=
\{E_a\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}
\\&&=
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme V-SuperHyperConnected}}
\\&&=\{V^{EXTERNAL}_a\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_b\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme V-SuperHyperConnected SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}

\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperBipartite $ESHB:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperConnected. The latter is straightforward. Then there's no at least one SuperHyperConnected. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperConnected could be applied. There are only two SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest SuperHyperConnected taken from a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-of-SuperHyperPart SuperHyperEdges are attained in any solution
 \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM21a}
In the Extreme Figure \eqref{136NSHG21a}, 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{136NSHG21a}, is the Extreme  SuperHyperConnected.
\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected}}
\\&&=
\{E_a\in E_{{P_i}^{ESHG:(V,E)}},
\\&&
~\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}
\\&&=
z^{\min|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|}
\\&&
\text{where}~
\forall {P_i}^{ESHG:(V,E)}, ~|{P_i}^{ESHG:(V,E)}|
\\&&
=\min_{i}|{P_i}^{ESHG:(V,E)}\in{P}^{ESHG:(V,E)}|\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme V-SuperHyperConnected}}
\\&&=\{V^{EXTERNAL}_a\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},V^{EXTERNAL}_b\in V^{EXTERNAL}_{{P_i}^{ESHG:(V,E)}},~i\neq j\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme V-SuperHyperConnected SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{Extreme Cardinality}}}=(\sum_{i=|{P}^{ESHG:(V,E)}|} (|{P_i}^{ESHG:(V,E)}|\text{choose}~2)=z^2.
\end{eqnarray*}

\end{proposition}
\begin{proof}
 Let
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest SuperHyperConnected taken from a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$  There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperConnected. The latter is straightforward.  Then there's no at least one SuperHyperConnected. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperConnected could be applied. There are only $z'$ SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the  
   \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-of-SuperHyperPart SuperHyperEdges are attained in any solution
    \begin{eqnarray*}
 && P:
 \\&&
 V^{EXTERNAL}_1,E_1,
 \\&&V^{EXTERNAL}_2,E_2
 \end{eqnarray*}
 is a longest path taken from a connected   Extreme SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, 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{136NSHG22a}, is the  Extreme SuperHyperConnected.

\end{example}
\begin{proposition}
Assume a connected Extreme SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected}}=\{E_i\in E^{*}_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme SuperHyperConnected SuperHyperPolynomial}}
\\&&=|i~|~E_i\in {|{E^{*}}_{ESHG:(V,E)}|}_{\text{Extreme Cardinality}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme V-SuperHyperConnected}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{Extreme V-SuperHyperConnected SuperHyperPolynomial}}=
z.
\end{eqnarray*}
\end{proposition}
\begin{proof}
 Let
 \begin{eqnarray*}
 && P:{V^{EXTERNAL}}_i,{E^{*}}_i,CENTER,E_j.
 \end{eqnarray*}
  is a longest SuperHyperConnected taken from a connected Extreme SuperHyperWheel $ESHW:(V,E).$
 There's a new way to redefine as
\begin{eqnarray*}
&&
V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z
\equiv
\\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z.
\end{eqnarray*}
The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperConnected. The latter is straightforward. Then there's at least one SuperHyperConnected. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperConnected could be applied. The unique embedded SuperHyperConnected proposes some longest SuperHyperConnected excerpt from some representatives. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM23a}
In the Extreme Figure \eqref{136NSHG23a}, 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{136NSHG23a}, is the  Extreme SuperHyperConnected.
            \end{example}

 

\section{Background}
There are some scientific researches covering the topic of this research. In what follows, there are some discussion and literature reviews about them date back on February 19, 2023.
\\
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 ISO 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 ISO 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.
\\
The seminal paper and groundbreaking article is titled ``Some Super Hyper Degrees and Co-Super Hyper Degrees on Neutrosophic Super Hyper Graphs And Super Hyper Graphs Alongside Applications in Cancer’s Treatments'' in \textbf{Ref.} \cite{HG133} by Henry Garrett (2023). In this research article, a novel approach is implemented on SuperHyperGraph and neutrosophic SuperHyperGraph based on fundamental notions and using vital tools in Cancer’s Treatments. It's published in prestigious and fancy journal is entitled “Journal of Mathematical Techniques and Computational Mathematics(JMTCM)” with ISO abbreviation ``J Math Techniques Comput Math'' in volume 2 and issue 1 with pages 35-47. 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),  ``Extreme SuperHyperClique as the Firm Scheme of Confrontation under Cancer’s Recognition as the Model in The Setting of (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG6} by Henry Garrett (2022), ``Uncertainty On The Act And Effect Of Cancer Alongside The Foggy Positions Of Cells Toward Neutrosophic Failed SuperHyperClique inside Neutrosophic SuperHyperGraphs Titled Cancer’s Recognition'' in \textbf{Ref.} \cite{HG7} by Henry Garrett (2022), ``Neutrosophic Version Of Separates Groups Of Cells In Cancer’s Recognition On Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG8} by Henry Garrett (2022), ``The Shift Paradigm To Classify Separately The Cells and Affected Cells Toward The Totality Under Cancer’s Recognition By New Multiple Definitions On the Sets Polynomials Alongside Numbers In The (Neutrosophic) SuperHyperMatching Theory Based on SuperHyperGraph and Neutrosophic SuperHyperGraph'' in \textbf{Ref.} \cite{HG9} by Henry Garrett (2022), ``Breaking the Continuity and Uniformity of Cancer In The Worst Case of Full Connections With Extreme Failed SuperHyperClique In Cancer’s Recognition Applied in (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG10} 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{HG11} 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{HG12} by Henry Garrett (2022), ``(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG13} 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{HG14} 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{HG15} by Henry Garrett (2022),  ``(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well- SuperHyperModelled (Neutrosophic) SuperHyperGraphs
'' in \textbf{Ref.} \cite{HG16} 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{HG17} 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{HG18} by Henry Garrett (2022),``(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances'' in \textbf{Ref.} \cite{HG19} 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{HG20} by Henry Garrett (2022), ``SuperHyperGirth on SuperHyperGraph and Neutrosophic SuperHyperGraph With SuperHyperModeling of Cancer’s Recognitions'' in \textbf{Ref.} \cite{HG21} by Henry Garrett (2022), ``Some SuperHyperDegrees and Co-SuperHyperDegrees on Neutrosophic SuperHyperGraphs and SuperHyperGraphs Alongside Applications in Cancer’s Treatments'' in \textbf{Ref.} \cite{HG22} by Henry Garrett (2022), ``SuperHyperDominating and SuperHyperResolving on Neutrosophic SuperHyperGraphs And Their Directions in Game Theory and Neutrosophic SuperHyperClasses'' in \textbf{Ref.} \cite{HG23} by Henry Garrett (2022), ``SuperHyperMatching By (R-)Definitions And Polynomials To Monitor Cancer’s Recognition In Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG24} by Henry Garrett (2023), ``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'' in \textbf{Ref.} \cite{HG25} by Henry Garrett (2023), ``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{HG26} 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{HG27} by Henry Garrett (2023), ``Perfect Directions Toward Idealism in Cancer's Neutrosophic Recognition Forwarding Neutrosophic SuperHyperClique on Neutrosophic SuperHyperGraphs'' in \textbf{Ref.} \cite{HG28} 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{HG29} 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{HG30} by Henry Garrett (2023), ``Using the Tool As (Neutrosophic) Failed SuperHyperStable To SuperHyperModel Cancer's Recognition Titled (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG31} 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{HG32} by Henry Garrett (2023), ``(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well-SuperHyperModelled (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG33} 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{HG34} by Henry Garrett (2022),  ``(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG35} by Henry Garrett (2022), ``Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs'' in \textbf{Ref.} \cite{HG36} by Henry Garrett (2022), ``Basic Neutrosophic Notions Concerning SuperHyperDominating and Neutrosophic SuperHyperResolving in SuperHyperGraph'' in \textbf{Ref.} \cite{HG37} 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{HG38} by Henry Garrett (2022), and \cite{HG94,HG942,HG95,HG952,HG96,HG962,HG97,HG972,HG98,HG982,HG106,HG107,HG111,HG112,HG115,HG116,HG120,HG121,HG122,HG123,HG124,HG125,HG126,HG127,HG128,HG129,HG130,HG131,HG132,HG133,HG134,HG135,HG136,HG137,HG138,HG139,HG140,HG141,HG142,HG143,HG144,HG145,HG146,HG147,HG148,HG149,HG150,HG151,HG152,HG153,HG154,HG155,HG156,HG157,HG158,HG159,HG160,HG161,HG162,HG163,HG164,HG165,HG166,HG167,HG168,HG169,HG170}, there are some endeavors to formalize the basic SuperHyperNotions about neutrosophic SuperHyperGraph and SuperHyperGraph.    
\\
Some scientific studies and scientific researches about neutrosophic graphs, are proposed as book in \textbf{Ref.} \cite{HG39} by Henry Garrett (2022) which is indexed by Google Scholar and has more than 3230 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 scientific studies and scientific researches about neutrosophic graphs, are proposed as book in \textbf{Ref.} \cite{HG40} by Henry Garrett (2022) which is indexed by Google Scholar and has more than 4117 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.
\\
See the seminal scientific researches \cite{HG1,HG2,HG3}. The formalization of the notions on the framework of Extreme K-Tree-Decomposition In SuperHyperGraphs, Neutrosophic K-Tree-Decomposition In SuperHyperGraphs theory, and (Neutrosophic) SuperHyperGraphs theory at \cite{HG4,HG5,HG6,HG7,HG8,HG9,HG10,HG11,HG12,HG13,HG14,HG15,HG16,HG17,HG18,HG19,HG20,HG21,HG22,HG23,HG24,HG25,HG26,HG27,HG28,HG29,HG30,HG31,HG32,HG33,HG34,HG35,HG36,HG37,HG38,HG94,HG942,HG95,HG952,HG96,HG962,HG97,HG972,HG98,HG982,HG106,HG107,HG111,HG112,HG115,HG116,HG120,HG121,HG122,HG123,HG124,HG125,HG126,HG127,HG128,HG129,HG130,HG131,HG132,HG133,HG134,HG135,HG136,HG137,HG138,HG139,HG140,HG141,HG142,HG143,HG144,HG145,HG146,HG147,HG148,HG149,HG150,HG151,HG152,HG153,HG154,HG155,HG156,HG157,HG158,HG159,HG160,HG161,HG162,HG163,HG164,HG165,HG166,HG167,HG168,HG169,HG170}. Two popular scientific research books in Scribd in the terms of high readers, 3230 and 4117 respectively,  on neutrosophic science is on \cite{HG39,HG40}.

 

 

 

--

\begin{thebibliography}{595}

\bibitem{HG1} Henry Garrett, ``\textit{Properties of SuperHyperGraph and Neutrosophic SuperHyperGraph}'', Neutrosophic Sets and Systems 49 (2022) 531-561 (doi: 10.5281/zenodo.6456413).  (http://fs.unm.edu/NSS/NeutrosophicSuperHyperGraph34.pdf).  (https://digitalrepository.unm.edu/nss\_journal/vol49/iss1/34).

\bibitem{HG2} Henry Garrett, ``\textit{Neutrosophic Co-degree and Neutrosophic Degree alongside Chromatic Numbers in the Setting of Some Classes Related to Neutrosophic Hypergraphs}'', J Curr Trends Comp Sci Res 1(1) (2022) 06-14.

\bibitem{HG3} Henry Garrett, “Super Hyper Dominating and Super Hyper Resolving on Neutrosophic Super Hyper Graphs and Their Directions in Game Theory and Neutrosophic Super Hyper Classes”, J Math Techniques Comput Math 1(3) (2022) 242-263. (doi: 10.33140/JMTCM.01.03.09)

\bibitem{HG133} Henry Garrett, “Some Super Hyper Degrees and Co-Super Hyper Degrees on Neutrosophic Super Hyper Graphs And Super Hyper Graphs Alongside Applications in Cancer’s Treatments”, J Math Techniques Comput Math 2(1) (2023) 35-47. (https://www.opastpublishers.com/open-access-articles/some-super-hyper-degrees-and-cosuper-hyper-degrees-on-neutrosophic-super-hyper-graphs-and-super-hyper-graphs-alongside-a.pdf)

\bibitem{HG4}
Garrett, Henry. ``\textit{0039 | Closing Numbers and Super-Closing Numbers as (Dual)Resolving and (Dual)Coloring alongside (Dual)Dominating in (Neutrosophic)n-SuperHyperGraph.}'' CERN European Organization for Nuclear Research - Zenodo, Nov. 2022. CERN European Organization for Nuclear Research, https://doi.org/10.5281/zenodo.6319942.
https://oa.mg/work/10.5281/zenodo.6319942

\bibitem{HG5}
Garrett, Henry. ``\textit{0049 | (Failed)1-Zero-Forcing Number in Neutrosophic Graphs.}'' CERN European Organization for Nuclear Research - Zenodo, Feb. 2022. CERN European Organization for Nuclear Research, https://doi.org/10.13140/rg.2.2.35241.26724.
https://oa.mg/work/10.13140/rg.2.2.35241.26724

\bibitem{HG6}
Henry Garrett, ``\textit{Extreme SuperHyperClique as the Firm Scheme of Confrontation under Cancer’s Recognition as the Model in The Setting of (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010308 (doi: 10.20944/preprints202301.0308.v1).

\bibitem{HG7}
Henry Garrett, ``\textit{Uncertainty On The Act And Effect Of Cancer Alongside The Foggy Positions Of Cells Toward Neutrosophic Failed SuperHyperClique inside Neutrosophic SuperHyperGraphs Titled Cancer’s Recognition}'', Preprints 2023, 2023010282 (doi: 10.20944/preprints202301.0282.v1).

\bibitem{HG8}
Henry Garrett, ``\textit{Neutrosophic Version Of Separates Groups Of Cells In Cancer’s Recognition On Neutrosophic SuperHyperGraphs}'', Preprints 2023, 2023010267 (doi: 10.20944/preprints202301.0267.v1).

\bibitem{HG9}
Henry Garrett, ``\textit{The Shift Paradigm To Classify Separately The Cells and Affected Cells Toward The Totality Under Cancer’s Recognition By New Multiple Definitions On the Sets Polynomials Alongside Numbers In The (Neutrosophic) SuperHyperMatching Theory Based on SuperHyperGraph and Neutrosophic SuperHyperGraph}'', Preprints 2023, 2023010265 (doi: 10.20944/preprints202301.0265.v1).

\bibitem{HG10}
Henry Garrett, ``\textit{Breaking the Continuity and Uniformity of Cancer In The Worst Case of Full Connections With Extreme Failed SuperHyperClique In Cancer’s Recognition Applied in (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010262,(doi: 10.20944/preprints202301.0262.v1).

\bibitem{HG11}
Henry Garrett, ``\textit{Neutrosophic Failed SuperHyperStable as the Survivors on the Cancer’s Neutrosophic Recognition Based on Uncertainty to All Modes in Neutrosophic SuperHyperGraphs}'', Preprints 2023, 2023010240 (doi: 10.20944/preprints202301.0240.v1).

\bibitem{HG12}
Henry Garrett, ``\textit{Extremism of the Attacked Body Under the Cancer's Circumstances Where Cancer's Recognition Titled (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010224, (doi: 10.20944/preprints202301.0224.v1).

\bibitem{HG13}
Henry Garrett, ``\textit{(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010105 (doi: 10.20944/preprints202301.0105.v1).

\bibitem{HG14}
Henry Garrett, ``\textit{Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints}'', Preprints 2023, 2023010088 (doi: 10.20944/preprints202301.0088.v1).

\bibitem{HG15}
Henry Garrett, ``\textit{Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond}'', Preprints 2023, 2023010044

\bibitem{HG16}
Henry Garrett, ``\textit{(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well- SuperHyperModelled (Neutrosophic) SuperHyperGraphs}'', Preprints 2023, 2023010043 (doi: 10.20944/preprints202301.0043.v1).

\bibitem{HG17}  Henry Garrett, \textit{``Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs''}, Preprints 2023, 2023010105 (doi: 10.20944/preprints202301.0105.v1).

\bibitem{HG18}  Henry Garrett, \textit{``Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints''}, Preprints 2023, 2023010088 (doi: 10.20944/preprints202301.0088.v1).

\bibitem{HG19}  Henry Garrett, \textit{``(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances''}, Preprints 2022, 2022120549 (doi: 10.20944/preprints202212.0549.v1).

 \bibitem{HG20} Henry Garrett, ``\textit{(Neutrosophic) SuperHyperAlliances With SuperHyperDefensive and SuperHyperOffensive Type-SuperHyperSet On (Neutrosophic) SuperHyperGraph With (Neutrosophic) SuperHyperModeling of Cancer’s Recognitions And Related (Neutrosophic) SuperHyperClasses}'', Preprints 2022, 2022120540 (doi: 10.20944/preprints202212.0540.v1).

\bibitem{HG21} Henry Garrett, ``\textit{SuperHyperGirth on SuperHyperGraph and Neutrosophic SuperHyperGraph With SuperHyperModeling of Cancer’s Recognitions}'', Preprints 2022, 2022120500 (doi: 10.20944/preprints202212.0500.v1).

\bibitem{HG22} Henry Garrett, ``\textit{Some SuperHyperDegrees and Co-SuperHyperDegrees on Neutrosophic SuperHyperGraphs and SuperHyperGraphs Alongside Applications in Cancer’s Treatments}'', Preprints 2022, 2022120324 (doi: 10.20944/preprints202212.0324.v1).

\bibitem{HG23} Henry Garrett, ``\textit{SuperHyperDominating and SuperHyperResolving on Neutrosophic SuperHyperGraphs And Their Directions in Game Theory and Neutrosophic SuperHyperClasses}'', Preprints 2022, 2022110576 (doi: 10.20944/preprints202211.0576.v1).

 

 

\bibitem{HG170}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By SuperHyperK-Tree-Decomposition As Hyper k-Actions On Super Patterns”, ResearchGate 2023, (doi: 10.13140/RG.2.2.19944.14086).
\bibitem{HG169}
Henry Garrett, “New Ideas On Super Harmony By Hyper k-Function Of SuperHyperK-Tree-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.23299.58404).
\bibitem{HG168}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By SuperHyperK-Number As Hyper k-Partition On Super Layers”, ResearchGate 2023, (doi: 10.13140/RG.2.2.33103.76968).
\bibitem{HG167}
Henry Garrett, “New Ideas On Super Gradient By Hyper k-Class Of SuperHyperK-Number In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.23037.44003).
\bibitem{HG166}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By SuperHyperOrder As Hyper Enumerations On Super Landmarks”, ResearchGate 2023, (doi: 10.13140/RG.2.2.35646.56641).
\bibitem{HG165}
Henry Garrett, “New Ideas On Super Analogous By Hyper Visions Of SuperHyperOrder In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.18030.48967).
\bibitem{HG164}
Henry Garrett, “New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By SuperHyperColoring As Hyper Categories On Super Neighbors”, ResearchGate 2023, (doi: 10.13140/RG.2.2.13973.81121).\bibitem{HG163}
Henry Garrett, “New Ideas On Super Relations By Hyper Identifications Of SuperHyperColoring In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.34106.47047).

\bibitem{HG163} Henry Garrett, “New Ideas On Super Relations By Hyper Identifications Of SuperHyperColoring In Cancer’s Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.34106.47047).

\bibitem{HG162} Henry Garrett, “New Ideas On Super Contradiction By Hyper Detection of SuperHyperDefensive In Cancer’s Neutrosophic Recognition and Neutrosophic SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.13397.09446).

\bibitem{HG161}
Henry Garrett, “New Ideas In Cancer’s Recognition And Neutrosophic SuperHyperGraph By SuperHyperDimension As Hyper Distinguishing On Super Distances”, ResearchGate 2023, (doi: 10.13140/RG.2.2.31956.88961).

\bibitem{HG160} Henry Garrett, “New Ideas On Super Locations By Hyper Differing Of SuperHyperDimension In Cancer’s Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.15179.67361).

\bibitem{HG159}
Henry Garrett, “New Ideas In Cancer’s Recognition And Neutrosophic SuperHyperGraph By SuperHyperDominating As Hyper Closing On Super Messy”, ResearchGate 2023, (doi: 10.13140/RG.2.2.21510.45125).
\bibitem{HG158}
Henry Garrett, “New Ideas On Super Missing By Hyper Searching Of SuperHyperDominating In Cancer’s Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.13121.84321).
\bibitem{HG157}
Henry Garrett, “New Ideas In Cancer’s Recognition And Neutrosophic SuperHyperGraph By SuperHyperConnected As Hyper Group On Super Surge”, ResearchGate 2023, (doi: 10.13140/RG.2.2.11758.69441).
\bibitem{HG156}
Henry Garrett, “New Ideas On Super Outbreak By Hyper Collections Of SuperHyperConnected In Cancer’s Recognition With (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.31891.35367).
\bibitem{HG155}
Henry Garrett, “New Ideas In Cancer’s Recognition And Neutrosophic SuperHyperGraph By SuperHyperTotal As Hyper Covering On Super Infections”, ResearchGate 2023, (doi: 10.13140/RG.2.2.19360.87048).
\bibitem{HG154}
Henry Garrett, “New Ideas On Super Extremism By Hyper Treatments Of SuperHyperTotal In Cancer’s Recognition with (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.32363.21286).
\bibitem{HG153}
  Henry Garrett, “New Ideas On Super Isolation By Hyper Perfectness Of SuperHyperPerfect In Cancer’s Recognition with (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.23266.81602).
\bibitem{HG152}
Henry Garrett, “New Ideas In Cancer’s Recognition And Neutrosophic SuperHyperGraph By SuperHyperPerfect As Hyper Idealism On Super Vacancy”, ResearchGate 2023, (doi: 10.13140/RG.2.2.19911.37285).
\bibitem{HG151}
Henry Garrett, “New Ideas In Cancer’s Recognition And Neutrosophic SuperHyperGraph By SuperHyperJoin As Hyper Separations On Super Sorts”, ResearchGate 2023, (doi: 10.13140/RG.2.2.11050.90569).
\bibitem{HG150}
Henry Garrett, “New Ideas On Super connections By Hyper disconnections Of SuperHyperJoin In Cancer’s Recognition with (Neutrosophic) SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.17761.79206).
\bibitem{HG149}
Henry Garrett, “New Ideas On Super Mixed-Devastations By Hyper Decisions Of SuperHyperDuality In Cancer’s Neutrosophic Recognition and Neutrosophic SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.34953.52320).
\bibitem{HG148}
Henry Garrett, “New Ideas In Cancer’s Recognition As (Neutrosophic) SuperHyperGraph By SuperHyperDuality As Hyper Imaginations On Super Mixed-Illustrations”, ResearchGate 2023, (doi: 10.13140/RG.2.2.33275.80161).
\bibitem{HG147}
Henry Garrett, “New Ideas In Cancer’s Recognition As (Neutrosophic) SuperHyperGraph By Path SuperHyperColoring As Hyper Correction On Super Faults”, ResearchGate 2023, (doi: 10.13140/RG.2.2.30182.50241).
\bibitem{HG146}
Henry Garrett, “New Ideas On Super Reflections By Hyper Rotations Of Path SuperHyperColoring In Cancer’s Neutrosophic Recognition and Neutrosophic SuperHyperGraph”, ResearchGate 2023, (doi: 10.13140/RG.2.2.33459.30243).
\bibitem{HG145}
Henry Garrett, “New Ideas As Hyper Deformations On Super Chains In Cancer’s Neutrosophic Recognition and Neutrosophic SuperHyperGraph By SuperHyperDensity”, ResearchGate 2023, (doi: 10.13140/RG.2.2.13444.60806).
\bibitem{HG144}
Henry Garrett, “New Ideas As Hyper Ignorance By SuperHyperDensity On Super Resistances In Cancer’s Neutrosophic Recognition and Neutrosophic SuperHyperGraph”, ResearchGate 2023, (doi:10.13140/RG.2.2.16800.05123).
\bibitem{HG143}
Henry Garrett, “New Research On Cancer’s Recognition and Neutrosophic SuperHyperGraph By Eulerian SuperHyperCycles and Hamiltonian Sets As Hyper Covering Versus Super separations-VI”, ResearchGate 2023, (doi: 10.13140/RG.2.2.29913.80482).
\bibitem{HG142}
Henry Garrett, “New Research On Cancer’s Recognition and Neutrosophic SuperHyperGraph By Eulerian SuperHyperCycles and Hamiltonian Sets As Hyper Covering Versus Super separations-V”, ResearchGate 2023, (doi: 10.13140/RG.2.2.33269.24809).                
\bibitem{HG141}
Henry Garrett, “New Research On Cancer’s Recognition and Neutrosophic SuperHyperGraph By Eulerian SuperHyperCycles and Hamiltonian Sets As Hyper Covering Versus Super separations-IV”, ResearchGate 2023, (doi: 10.13140/RG.2.2.34946.96960).
\bibitem{HG140}
Henry Garrett, “A Research On Cancer’s Recognition and Neutrosophic SuperHyperGraph By Eulerian SuperHyperCycles and Hamiltonian Sets As Hyper Covering Versus Super separations-III”, ResearchGate 2023, (doi: 10.13140/RG.2.2.14814.31040).
\bibitem{HG139}
Henry Garrett, “A Research On Cancer’s Recognition and Neutrosophic SuperHyperGraph By Eulerian SuperHyperCycles and Hamiltonian Sets As Hyper Covering Versus Super separations-II”, ResearchGate 2023, (doi: 10.13140/RG.2.2.15653.17125).
\bibitem{HG138}
Henry Garrett, “A Research On Cancer’s Recognition and Neutrosophic SuperHyperGraph By Eulerian SuperHyperCycles and Hamiltonian Sets As Hyper Covering Versus Super separations-I”, ResearchGate 2023, (doi: 10.13140/RG.2.2.25719.50089).
\bibitem{HG137}
Henry Garrett, “New Ideas On Super Disruptions In Cancer’s Extreme Recognition As Neutrosophic SuperHyperGraph By Hyper Plans Called SuperHyperConnectivities”, ResearchGate 2023, (doi: 10.13140/RG.2.2.29441.94562).
\bibitem{HG136}
Henry Garrett, “Cancer’s Neutrosophic Recognition As Neutrosophic SuperHyperGraph By SuperHyperConnectivities As Hyper Diagnosis On Super Mechanism”, ResearchGate 2023, (doi: 10.13140/RG.2.2.17252.24968).
\bibitem{HG135}
  Henry Garrett,“Cancer’s Recognition and (Neutrosophic) SuperHyperGraph By the Criteria of Eulerian and Hamiltonian Type-Sets As Hyper Modified Cycles On Super Mess”, ResearchGate 2023,(doi: 10.13140/RG.2.2.16652.59525).
\bibitem{HG134}
Henry Garrett,“Eulerian and Hamiltonian In Cancer’s Neutrosophic Recognition and Neutrosophic SuperHyperGraph On Super Interactions By Hyper Extensions of Cycles”, ResearchGate 2023,(doi: 10.13140/RG.2.2.34583.24485).

\bibitem{HG133}  Henry Garrett, “Some Super Hyper Degrees and Co-Super Hyper Degrees on Neutrosophic Super Hyper Graphs And Super Hyper Graphs Alongside Applications in Cancer’s Treatments”, J Math Techniques Comput Math 2(1) (2023) 35-47.

\bibitem{HG132} Henry Garrett,“SuperHyperGirth Type-Results on extreme SuperHyperGirth theory and (Neutrosophic) SuperHyperGraphs Toward Cancer’s extreme Recognition”, Preprints 2023, 2023010396 (doi: 10.20944/preprints202301.0396.v1).

\bibitem{HG131} Henry Garrett,“Neutrosophic SuperHyperGraphs Warns Hyper Landmark of neutrosophic SuperHyperGirth In Super Type-Versions of Cancer’s neutrosophic Recognition”, Preprints 2023, 2023010395 (doi: 10.20944/preprints202301.0395.v1).

\bibitem{HG130} Henry Garrett,“The Constructions of (Neutrosophic) SuperHyperGraphs on the Cancer’s Recognition in The Confrontation With Super Attacks In Hyper Situations By Implementing (Neutrosophic) 1-Failed SuperHyperForcing in The Technical Approaches to Neutralize SuperHyperViews”, ResearchGate 2023, (doi: 10.13140/RG.2.2.26240.51204).

\bibitem{HG129} Henry Garrett,“(Neutrosophic) 1-Failed SuperHyperForcing As the Entrepreneurs on Cancer’s Recognitions To Fail Forcing Style As the Super Classes With Hyper Effects In The Background of the Framework is So-Called (Neutrosophic) SuperHyperGraphs”, ResearchGate 2023, (doi: 10.13140/RG.2.2.12818.73925).

\bibitem{HG128} Henry Garrett,“Super Actions On The Types of Hyper Levels In The Sensible Neutrosophic SuperHyperGirth On Cancer’s Neutrosophic Recognition and Neutrosophic SuperHyperGraph”, ResearchGate 2023,(doi: 10.13140/RG.2.2.26836.88960).

\bibitem{HG127} Henry Garrett,“SuperHyperGirth Approaches on the Super Challenges on the Cancer’s Recognition In the Hyper Model of (Neutrosophic) SuperHyperGraph”, ResearchGate 2023,(doi: 10.13140/RG.2.2.36745.93289).

\bibitem{HG126}  Henry Garrett,“Extreme SuperHyperClique as the Firm Scheme of Confrontation under Cancer’s Recognition as the Model in The Setting of (Neutrosophic) SuperHyperGraphs”, Preprints 2023, 2023010308 (doi: 10.20944/preprints202301.0308.v1).

\bibitem{HG125} Henry Garrett,“Uncertainty On The Act And Effect Of Cancer Alongside The Foggy Positions Of Cells Toward Neutrosophic Failed SuperHyperClique inside Neutrosophic SuperHyperGraphs Titled Cancer’s Recognition”, Preprints 2023, 2023010282 (doi: 10.20944/preprints202301.0282.v1).

\bibitem{HG124}  Henry Garrett,“Neutrosophic Version Of Separates Groups Of Cells In Cancer’s Recognition On Neutrosophic SuperHyperGraphs”, Preprints 2023, 2023010267 (doi: 10.20944/preprints202301.0267.v1).).

\bibitem{HG123}  Henry Garrett, “The Shift Paradigm To Classify Separately The Cells and Affected Cells Toward The Totality Under Cancer’s Recognition By New Multiple Definitions On the Sets Polynomials Alongside Numbers In The (Neutrosophic) SuperHyperMatching Theory Based on SuperHyperGraph and Neutrosophic SuperHyperGraph”, Preprints 2023, 2023010265 (doi: 10.20944/preprints202301.0265.v1).

\bibitem{HG122} Henry Garrett,“Breaking the Continuity and Uniformity of Cancer In The Worst Case of Full Connections With Extreme Failed SuperHyperClique In Cancer’s Recognition Applied in (Neutrosophic) SuperHyperGraphs”, Preprints 2023, 2023010262,(doi: 10.20944/preprints202301.0262.v1).

\bibitem{HG121}  Henry Garrett, “Neutrosophic Failed SuperHyperStable as the Survivors on the Cancer’s Neutrosophic Recognition Based on Uncertainty to All Modes in Neutrosophic SuperHyperGraphs”, Preprints 2023, 2023010240 (doi: 10.20944/preprints202301.0240.v1).

\bibitem{HG120} Henry Garrett, “Extremism of the Attacked Body Under the Cancer’s Circumstances Where Cancer’s Recognition Titled (Neutrosophic) SuperHyperGraphs”, Preprints 2023, 2023010224, (doi: 10.20944/preprints202301.0224.v1).

\bibitem{HG24} Henry Garrett,``\textit{SuperHyperMatching By (R-)Definitions And Polynomials To Monitor Cancer’s Recognition In Neutrosophic SuperHyperGraphs}'', ResearchGate 2023,(doi: 10.13140/RG.2.2.35061.65767).

\bibitem{HG25} Henry Garrett,``\textit{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}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.18494.15680).

\bibitem{HG26} Henry Garrett,``\textit{Extreme Failed SuperHyperClique Decides the Failures on the Cancer's Recognition in the Perfect Connections of Cancer's Attacks By SuperHyperModels Named (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.32530.73922).

\bibitem{HG27} Henry Garrett,``\textit{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}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.15897.70243).

 

\bibitem{HG116}  Henry Garrett,“Extreme Failed SuperHyperClique Decides the Failures on the Cancer’s Recognition in the Perfect Connections of Cancer’s Attacks By SuperHyperModels Named (Neutrosophic) SuperHyperGraphs”, ResearchGate 2023, (doi: 10.13140/RG.2.2.32530.73922).

\bibitem{HG115}  Henry Garrett, “(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs”, Preprints 2023, 2023010105 (doi: 10.20944/preprints202301.0105.v1).

\bibitem{HG28} Henry Garrett,``\textit{Perfect Directions Toward Idealism in Cancer's Neutrosophic Recognition Forwarding Neutrosophic SuperHyperClique on Neutrosophic SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.30092.80004).

\bibitem{HG29} Henry Garrett,``\textit{Demonstrating Complete Connections in Every Embedded Regions and Sub-Regions in the Terms of Cancer's Recognition and (Neutrosophic) SuperHyperGraphs With (Neutrosophic) SuperHyperClique}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.23172.19849).

 

\bibitem{HG112}  Henry Garrett, “Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs”, Preprints 2023, 2023010105 (doi: 10.20944/preprints202301.0105.v1).

\bibitem{HG111} Henry Garrett, “Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints”, Preprints 2023, 2023010088 (doi: 10.20944/preprints202301.0088.v1).

 

\bibitem{HG30} Henry Garrett,``\textit{Different Neutrosophic Types of Neutrosophic Regions titled neutrosophic Failed SuperHyperStable in Cancer’s Neutrosophic Recognition modeled in the Form of Neutrosophic SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.17385.36968).

 

\bibitem{HG107} Henry Garrett, “Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond”, Preprints 2023, 2023010044

\bibitem{HG106} Henry Garrett, “(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well- SuperHyperModelled (Neutrosophic) SuperHyperGraphs”, Preprints 2023, 2023010043 (doi: 10.20944/preprints202301.0043.v1).

\bibitem{HG31} Henry Garrett, ``\textit{Using the Tool As (Neutrosophic) Failed SuperHyperStable To SuperHyperModel Cancer's Recognition Titled (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.28945.92007).

\bibitem{HG32} Henry Garrett, ``\textit{Neutrosophic Messy-Style SuperHyperGraphs To Form Neutrosophic SuperHyperStable To Act on Cancer’s Neutrosophic Recognitions In Special ViewPoints}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.11447.80803).

\bibitem{HG33} Henry Garrett, ``\textit{(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well-SuperHyperModelled (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2023, (doi: 10.13140/RG.2.2.35774.77123).

\bibitem{HG34} Henry Garrett, ``\textit{Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond}'', ResearchGate 2022, (doi: 10.13140/RG.2.2.36141.77287).

\bibitem{HG35} Henry Garrett, ``\textit{(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2022, (doi: 10.13140/RG.2.2.29430.88642).

\bibitem{HG36} Henry Garrett, ``\textit{Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs}'', ResearchGate 2022, (doi: 10.13140/RG.2.2.11369.16487).

 

 

 

 

\bibitem{HG982}  Henry Garrett, “(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances”, Preprints 2022, 2022120549 (doi: 10.20944/preprints202212.0549.v1).

\bibitem{HG98}  Henry Garrett, “(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances”, ResearchGate 2022, (doi: 10.13140/RG.2.2.19380.94084).

\bibitem{HG972}  Henry Garrett, “(Neutrosophic) SuperHyperAlliances With SuperHyperDefensive and SuperHyperOffensive Type-SuperHyperSet On (Neutrosophic) SuperHyperGraph With (Neutrosophic) SuperHyperModeling of Cancer’s Recognitions And Related (Neutrosophic) SuperHyperClasses”, Preprints 2022, 2022120540 (doi: 10.20944/preprints202212.0540.v1).

\bibitem{HG97}  Henry Garrett, “(Neutrosophic) SuperHyperAlliances With SuperHyperDefensive and SuperHyperOffensive Type-SuperHyperSet On (Neutrosophic) SuperHyperGraph With (Neutrosophic) SuperHyperModeling of Cancer’s Recognitions And Related (Neutrosophic) SuperHyperClasses”, ResearchGate 2022, (doi: 10.13140/RG.2.2.14426.41923).

\bibitem{HG962} Henry Garrett, “SuperHyperGirth on SuperHyperGraph and Neutrosophic SuperHyperGraph With SuperHyperModeling of Cancer’s Recognitions”, Preprints 2022, 2022120500 (doi: 10.20944/preprints202212.0500.v1).

\bibitem{HG96} Henry Garrett, “SuperHyperGirth on SuperHyperGraph and Neutrosophic SuperHyperGraph With SuperHyperModeling of Cancer’s Recognitions”, ResearchGate 2022 (doi: 10.13140/RG.2.2.20993.12640).

\bibitem{HG952}  Henry Garrett,“Some SuperHyperDegrees and Co-SuperHyperDegrees on Neutrosophic SuperHyperGraphs and SuperHyperGraphs Alongside Applications in Cancer’s Treatments”, Preprints 2022, 2022120324 (doi: 10.20944/preprints202212.0324.v1).

\bibitem{HG95} Henry Garrett, “Some SuperHyperDegrees and Co-SuperHyperDegrees on Neutrosophic SuperHyperGraphs And SuperHyperGraphs Alongside Applications in Cancer’s Treatments”, ResearchGate 2022 (doi: 10.13140/RG.2.2.23123.04641).

\bibitem{HG942}  Henry Garrett, “SuperHyperDominating and SuperHyperResolving on Neutrosophic SuperHyperGraphs And Their Directions in Game Theory and Neutrosophic SuperHyperClasses”, Preprints 2022, 2022110576 (doi: 10.20944/preprints202211.0576.v1).

\bibitem{HG94} Henry Garrett, “SuperHyperDominating and SuperHyperResolving on Neutrosophic SuperHyperGraphs And Their Directions in Game Theory and Neutrosophic SuperHyperClasses”, ResearchGate 2022 (doi: 10.13140/RG.2.2.23324.56966).

\bibitem{HG37}  Henry Garrett, \textit{``Basic Neutrosophic Notions Concerning SuperHyperDominating and Neutrosophic SuperHyperResolving in SuperHyperGraph''}, ResearchGate 2022 (doi: 10.13140/RG.2.2.29173.86244).

\bibitem{HG38} Henry Garrett, ``\textit{Initial Material of Neutrosophic Preliminaries to Study Some Neutrosophic Notions Based on Neutrosophic SuperHyperEdge (NSHE) in Neutrosophic SuperHyperGraph (NSHG)}'', ResearchGate 2022 (doi: 10.13140/RG.2.2.25385.88160).

\bibitem{HG39} Henry Garrett, (2022). ``\textit{Beyond Neutrosophic Graphs}'', Ohio: E-publishing: Educational Publisher 1091 West 1st Ave Grandview Heights, Ohio 43212 United States. ISBN: 979-1-59973-725-6 (http://fs.unm.edu/BeyondNeutrosophicGraphs.pdf).
      
 \bibitem{HG40} Henry Garrett, (2022). ``\textit{Neutrosophic Duality}'', Florida: GLOBAL KNOWLEDGE - Publishing House 848 Brickell Ave Ste 950 Miami, Florida 33131 United States. ISBN: 978-1-59973-743-0 (http://fs.unm.edu/NeutrosophicDuality.pdf).  

\end{thebibliography}
\end{document}