New Ideas On Super Decompensation By Hyper Decompress Of Eulerian-Type-Cycle-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph
Authors/Creators
- 1. Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA
Description
260
Henry Garrett, “New Ideas On Super Decompensation By Hyper Decompress Of Eulerian-Type-Cycle-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph”, Zenodo 2023, (doi: 10.5281/zenodo.7904401).
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\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 }
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\textbf\newline{
New Ideas On Super Decompensation By Hyper Decompress Of Eulerian-Type-Cycle-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph
}
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Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA
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\section{ABSTRACT}
In this scientific research, (Different Neutrosophic Types of Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}). Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a {\tiny Eulerian-Type-Cycle-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-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} if the following expression is called Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} criteria holds
\begin{eqnarray*}
&&
\forall E_a\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
\text{the maximum number of SuperHyperEdges};
\end{eqnarray*}
Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} if the following expression is called Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} criteria holds
\begin{eqnarray*}
&&
\forall E_a\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
\text{the maximum number of SuperHyperEdges};
\end{eqnarray*}
and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$ Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} if the following expression is called Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} criteria holds
\begin{eqnarray*}
&&
\forall V_a\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
\text{the maximum number of SuperHyperEdges};
\end{eqnarray*}
Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} if the following expression is called Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} criteria holds
\begin{eqnarray*}
&&
\forall V_a\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
\text{the maximum number of SuperHyperEdges};
\end{eqnarray*}
and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$ Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. ((Neutrosophic) SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}; a Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}; an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}; and the Extreme power is corresponded to its Extreme coefficient; a Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}; and the Neutrosophic power is corresponded to its Neutrosophic coefficient; an Extreme V-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}; a Neutrosophic V-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}; an Extreme V-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}; and the Extreme power is corresponded to its Extreme coefficient; a Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}; and the Neutrosophic power is corresponded to its Neutrosophic coefficient. In this scientific research, new setting is introduced for new SuperHyperNotions, namely, a SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} and Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-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 $\delta-$SuperHyper{\tiny Eulerian-Type-Cycle-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 $\delta-$SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} . Since there's more ways to get type-results to make a SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} more understandable. For the sake of having Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, there's a need to ``redefine'' the notion of a ``SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} . It's redefined a Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} until the SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, then it's officially called a ``SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}'' but otherwise, it isn't a SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} . There are some instances about the clarifications for the main definition titled a ``SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} ''. These two examples get more scrutiny and discernment since there are characterized in the disciplinary ways of the SuperHyperClass based on a SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} . For the sake of having a Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, there's a need to ``redefine'' the notion of a ``Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}'' and a ``Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} are redefined to a ``Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} more understandable. Assume a Neutrosophic SuperHyperGraph. There are some Neutrosophic SuperHyperClasses if the intended Table holds. Thus SuperHyperPath, SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultiPartite, and SuperHyperWheel, are
``Neutrosophic SuperHyperPath'', ``Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}'', ``Neutrosophic SuperHyperStar'', ``Neutrosophic SuperHyperBipartite'', ``Neutrosophic SuperHyperMultiPartite'', and ``Neutrosophic SuperHyperWheel'' if the intended Table holds. A SuperHyperGraph has a ``Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}'' where it's the strongest [the maximum Neutrosophic value from all the SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} amid the maximum value amid all SuperHyperVertices from a SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} .] SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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(-/SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). The aim is to find either the longest SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} or the strongest SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in those Neutrosophic SuperHyperModels. For the longest SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, called SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and the strongest SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, called Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. There isn't any formation of any SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} but literarily, it's the deformation of any SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. It, literarily, deforms and it doesn't form. A basic familiarity with Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} theory, SuperHyperGraphs, and Neutrosophic SuperHyperGraphs theory are proposed.
\\ \vspace{4mm}
\textbf{Keywords:} Neutrosophic SuperHyperGraph, SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Cancer's Neutrosophic 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 ``an 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 ``an 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 an 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 `` SuperHyper{\tiny Eulerian-Type-Cycle-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 an 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 an Extreme SuperHyperPath (-/SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). The aim is to find either the optimal SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} or the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in those an Extreme SuperHyperModels. Some general results are introduced. Beyond that in SuperHyperStar, all possible an 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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. There isn't any formation of any SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} but literarily, it's the deformation of any SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 `` SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}'' and ``an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}'' on ``SuperHyperGraph'' and ``an 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 an 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, SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} and an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, are figured out in sections `` SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}'' and ``an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}''. In the sense of tackling on getting results and in {\tiny Eulerian-Type-Cycle-Decomposition} to make sense about continuing the research, the ideas of SuperHyperUniform and an 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 an 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 an Extreme SuperHyperGraph, in the sections ``Results on SuperHyperClasses'' and ``Results on an 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'',
`` SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}'', ``an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}'', ``Results on SuperHyperClasses'' and ``Results on an 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, `` SuperHyper{\tiny Eulerian-Type-Cycle-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{an 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 {\tiny Eulerian-Type-Cycle-Decomposition} 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 {\tiny Eulerian-Type-Cycle-Decomposition} 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 SuperHyper{\tiny Eulerian-Type-Cycle-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-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}} if the following expression is called \textbf{Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} criteria} holds
\begin{eqnarray*}
&&
\forall E_a\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
\text{the maximum number of SuperHyperEdges};
\end{eqnarray*}
\item[$(ii)$] \textbf{Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}} if the following expression is called \textbf{Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} criteria} holds
\begin{eqnarray*}
&&
\forall E_a\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
\text{the maximum number of SuperHyperEdges};
\end{eqnarray*}
and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(iii)$] \textbf{Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}} if the following expression is called \textbf{Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} criteria} holds
\begin{eqnarray*}
&&
\forall V_a\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
\text{the maximum number of SuperHyperEdges};
\end{eqnarray*}
\item[$(iv)$] \textbf{Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}} if the following expression is called \textbf{Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} criteria} holds
\begin{eqnarray*}
&&
\forall V_a\in C: C~\text{is}
\\&&
\text{a SuperHyperCycle and it has}
\\&&
\text{the maximum number of SuperHyperEdges};
\end{eqnarray*}
and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$
\item[$(v)$] \textbf{Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{definition}
\begin{definition}((Neutrosophic) SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii)$]
a \textbf{Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii)$]
an \textbf{Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}; and the Extreme power is corresponded to its Extreme coefficient;
\item[$(iv)$]
a \textbf{Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}; and the Neutrosophic power is corresponded to its Neutrosophic coefficient;
\item[$(v)$]
an \textbf{Extreme V-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi)$]
a \textbf{Neutrosophic V-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vii)$]
an \textbf{Extreme V-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}; and the Extreme power is corresponded to its Extreme coefficient;
\item[$(viii)$]
a \textbf{Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic re-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, Neutrosophic v-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and Neutrosophic rv-SuperHyper{\tiny Eulerian-Type-Cycle-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 conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}; and the Neutrosophic power is corresponded to its Neutrosophic coefficient.
\end{itemize}
\end{definition}
\begin{definition}((Extreme/Neutrosophic)$\delta-$SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}).\\
Assume a Neutrosophic SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Then
\begin{itemize}
\item[$(i)$] an \textbf{$\delta-$SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}} is a Neutrosophic kind of Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-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 $\delta-$SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}} is a Neutrosophic kind of Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 , SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. Since there's more ways to get type-results to make a Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} more Neutrosophicly understandable.
\\
For the sake of having a Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, there's a need to ``\textbf{redefine}'' the Neutrosophic notion of ``Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. It's redefined a \textbf{Neutrosophic SuperHyper{\tiny Eulerian-Type-Cycle-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{
an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(an Extreme event).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Any an Extreme k-subset of $A$ of $V$ is called \textbf{an Extreme k-event} and if $k=2,$ then an Extreme subset of $A$ of $V$ is called \textbf{an Extreme event}. The following expression is called \textbf{an Extreme probability} of $A.$
\begin{eqnarray}
E(A)=\sum_{a\in A}E(a).
\end{eqnarray}
\end{definition}
\begin{definition}(an Extreme Independent).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. $s$ an Extreme k-events $A_i,~i\in I$ is called \textbf{an Extreme s-independent} if the following expression is called \textbf{an Extreme s-independent criteria}
\begin{eqnarray*}
E(\cap_{i\in I}A_i)=\prod_{i\in I}P(A_i).
\end{eqnarray*}
And if $s=2,$ then an Extreme k-events of $A$ and $B$ is called \textbf{an Extreme independent}. The following expression is called \textbf{an Extreme independent criteria}
\begin{eqnarray}
E(A\cap B)=P(A)P(B).
\end{eqnarray}
\end{definition}
\begin{definition}(an Extreme Variable).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Any k-function {\tiny Eulerian-Type-Cycle-Decomposition} like $E$ is called \textbf{an Extreme k-Variable}. If $k=2$, then any 2-function {\tiny Eulerian-Type-Cycle-Decomposition} like $E$ is called \textbf{an Extreme Variable}.
\end{definition}
The notion of independent on an Extreme Variable is likewise.
\begin{definition}(an Extreme Expectation).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. A an Extreme k-Variable $E$ has a number is called \textbf{an Extreme Expectation} if the following expression is called \textbf{an Extreme Expectation criteria}
\begin{eqnarray*}
Ex(E)=\sum_{\alpha \in V}E(\alpha)P(\alpha).
\end{eqnarray*}
\end{definition}
\begin{definition}(an Extreme Crossing).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. A an Extreme number is called \textbf{an Extreme Crossing} if the following expression is called \textbf{an Extreme Crossing criteria}
\begin{eqnarray*}
Cr(S)=\min\{\text{Number of Crossing in a Plane Embedding of } S\}.
\end{eqnarray*}
\end{definition}
\begin{lemma}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Let $m$ and $n$ propose special {\tiny Eulerian-Type-Cycle-Decomposition}. Then with $m\geq 4n,$
\end{lemma}
\begin{proof}
Consider a planar embedding $G$ of $G$ with $cr(G)$ crossings. Let $S$ be a an Extreme random k-subset of $V$ obtained by choosing each SuperHyperVertex of $G$ an Extreme independently with probability {\tiny Eulerian-Type-Cycle-Decomposition} $p := 4n/m,$ and set $H:=G[S]$ and $H:=G[S].$
\\
Define random variables $X, Y, Z$ on $V$ as follows: $X$ is the an Extreme number of SuperHyperVertices, $Y$ the an Extreme number of SuperHyperEdges, and $Z$ the an Extreme number of crossings of $H.$ The trivial bound noted above, when applied to $H,$ yields the inequality $Z \geq cr(H) \geq Y-3X.$ By linearity of an Extreme Expectation,
$$E(Z) \geq E(Y )-3E(X).$$
Now $E(X) = pn,~E(Y ) = p^2m$ (each SuperHyperEdge having some SuperHyperEnds) and $E(Z) = p^4cr(G)$ (each crossing being defined by some SuperHyperVertices). Hence
$$p^4cr(G) \geq p^2m-3pn.$$
Dividing both sides by $p^4,$ we have:
\begin{eqnarray*}
cr(G)\geq \frac{pm-3n}{p^3}=\frac{n}{{(4n/m)}^3}=\frac{1}{64}{m^3}{n^2}.
\end{eqnarray*}
\end{proof}
\begin{theorem}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Let $P$ be a SuperHyperSet of $n$ points in the plane, and let $l$ be the an Extreme number of SuperHyperLines in the plane passing through at least $k+1$ of these points, where $1 \leq k \leq 2 \sqrt{2n}.$ Then $l<32n^2/k^3.$
\end{theorem}
\begin{proof}
Form a an Extreme SuperHyperGraph $G$ with SuperHyperVertex SuperHyperSet $P$ whose SuperHyperEdge are the segments between
conseNeighborive points on the SuperHyperLines which pass through at least $k+1$ points of $P.$ This
an Extreme SuperHyperGraph has at least $kl$ SuperHyperEdges and an Extreme crossing at most $l$ choose two. Thus either $kl < 4n,$ in which case $l < 4n/k \leq32n^2/k^3,$ or $l^2/2 > \text{l choose 2} \geq cr(G) \geq {(kl)}^3/64n^2$ by the an Extreme Crossing Lemma, and again $l < 32n^2/k^3.$
\end{proof}
\begin{theorem}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Let $P$ be a SuperHyperSet of $n$ points in the plane, and let $k$ be the number of pairs of points of $P$ at unit SuperHyperDistance. Then $k < 5n^{4/3}.$
\end{theorem}
\begin{proof}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Draw a SuperHyperUnit SuperHyperCircle around each SuperHyperPoint of $P.$ Let $n_i$ be the an Extreme number of these SuperHyperCircles passing through exactly $i$ points of $P.$ Then $\sum{i=0}^{n-1}n_i = n$ and $k = \frac{1}{2}\sum{i=0}^{n-1}in_i.$ Now form a an Extreme SuperHyperGraph $H$ with SuperHyperVertex SuperHyperSet $P$ whose SuperHyperEdges are the SuperHyperArcs between conseNeighborive SuperHyperPoints on the SuperHyperCircles that pass through at least three SuperHyperPoints
of $P.$ Then
\begin{eqnarray*}
e(H)=\sum_{i=3}^{n-1}in_i=2k-n_1-2n_2\geq2k-2n.
\end{eqnarray*}
Some SuperHyperPairs of SuperHyperVertices of $H$ might be joined by some parallel SuperHyperEdges. Delete from $H$ one of each SuperHyperPair of parallel SuperHyperEdges, so as to obtain a simple an Extreme SuperHyperGraph $G$ with $e(G) \geq k-n.$ Now $cr(G)\leq n(n-1)$ because $G$ is formed from at most n SuperHyperCircles, and any two SuperHyperCircles cross at most twice. Thus either $e(G) < 4n,$ in which case $k < 5n < 5n^{4/3},$ or $n^2 > n(n-1) \geq cr(G) \geq {(k-n)}^3/64n^2$ by the an Extreme Crossing Lemma, and $k<4n^{4/3} +n<5n^{4/3}.$
\end{proof}
\begin{proposition}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Let $X$ be a nonnegative an Extreme Variable and t a positive real number. Then
\begin{eqnarray*}
P(X\geq t) \leq \frac{E(X)}{t}.
\end{eqnarray*}
\end{proposition}
\begin{proof}
\begin{eqnarray*}
&& E(X)=\sum \{X(a)P(a):a\in V\} \geq \sum\{X(a)P(a):a\in V,X(a)\geq t\}
\\&&
\sum\{tP(a):a\in V,X(a)\geq t\}=t\sum\{P(a):a\in V,X(a)\geq t\}
\\&&
tP(X\geq t).
\end{eqnarray*}
Dividing the first and last members by $t$ yields the asserted inequality.
\end{proof}
\begin{corollary}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Let $X_n$ be a nonnegative integer-valued variable in a prob- ability {\tiny Eulerian-Type-Cycle-Decomposition} $(V_n,E_n), n\geq1.$ If $E(X_n)\rightarrow0$ as $n\rightarrow \infty,$ then $P(X_n =0)\rightarrow1$ as $n \rightarrow \infty.$
\end{corollary}
\begin{proof}
\end{proof}
\begin{theorem}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}.
A special SuperHyperGraph in $G_{n,p}$ almost surely has stability number at most $\lceil2p^{-1} \log n\rceil.$
\end{theorem}
\begin{proof}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}.
A special SuperHyperGraph in $G_{n,p}$ is up. Let $G\in \mathcal{G}_{n,p}$ and let $S$ be a given SuperHyperSet of $k+1$ SuperHyperVertices of $G,$ where $k\in \Bbb N.$ The probability that $S$ is a stable SuperHyperSet of $G$ is $(1-p)^{(k+1) \text{choose} 2},$ this being the probability that none of the $(k+1) \text{choose} 2$ pairs of SuperHyperVertices of $S$ is a SuperHyperEdge of the an Extreme SuperHyperGraph $G.$
\\
Let $A_S$ denote the event that $S$ is a stable SuperHyperSet of $G,$ and let $X_S$ denote the indicator an Extreme Variable for this an Extreme Event. By equation, we have
\begin{eqnarray*}
E(X_S) = P(X_S = 1) = P(A_S ) = (1-p)^{(k+1) \text{choose} 2}.
\end{eqnarray*}
Let $X$ be the number of stable SuperHyperSets of cardinality $k + 1$ in $G.$ Then
\begin{eqnarray*}
X = \sum\{X_S : S \subseteq V, |S| = k + 1\}
\end{eqnarray*}
and so, by those,
\begin{eqnarray*}
E(X)= \sum\{E(X_S): S\subseteq V, |S|=k+1\}= (\text{n choose k+1}) (1-p)^{(k+1) \text{choose} 2}.
\end{eqnarray*}
We bound the right-hand side by invoking two elementary inequalities:
\begin{eqnarray*}
(\text{n choose k+1})\leq \frac{n^{k+1}}{(k+1)!} \text{and} 1-p\leq e^{-p}.
\end{eqnarray*}
This yields the following upper bound on $E(X).$
\begin{eqnarray*}
E(X)\leq\frac{n^{k+1}e^{-p)(k+1) \text{choose} 2}}{(k+1)!}=\frac{ne^{-pk/2}^{k+1}}{(k+1)!}
\end{eqnarray*}
Suppose now that $k = \lceil2p^{-1} \log n\rceil.$ Then $k \geq 2p^{-1} \log n,$ so $ne^{-pk/2} \leq 1.$ Because $k$ grows at least as fast as the logarithm of $n,$ implies that $E(X) \rightarrow 0$ as $n \rightarrow \infty.$ Because $X$ is integer-valued and nonnegative, we deduce from Corollary that $P (X = 0) \rightarrow 1$ as $n \rightarrow \infty.$ Consequently, a an Extreme SuperHyperGraph in $\mathcal{G}_{n,p}$ almost surely has stability number at most $k.$
\end{proof}
\begin{definition}(an Extreme Variance).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. A an Extreme k-Variable $E$ has a number is called \textbf{an Extreme Variance} if the following expression is called \textbf{an Extreme Variance criteria}
\begin{eqnarray*}
Vx(E)=Ex({(X-Ex(X))}^2).
\end{eqnarray*}
\end{definition}
\begin{theorem}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Let $X$ be a an Extreme Variable and let $t$ be a positive real number. Then
\begin{eqnarray*}
E(|X-Ex(X)|\geq t)\leq \frac{V(X)}{t^2}.
\end{eqnarray*}
\end{theorem}
\begin{proof}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Let $X$ be a an Extreme Variable and let $t$ be a positive real number. Then
\begin{eqnarray*}
E(|X-Ex(X)|\geq t)=E{((X-Ex(X))}^2 \geq t^2)\leq \frac{Ex({(X-Ex(X))}^2)}{t^2}=\frac{V(X)}{t^2}.
\end{eqnarray*}
\end{proof}
\begin{corollary}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Let $X_n$ be a an Extreme Variable in a probability {\tiny Eulerian-Type-Cycle-Decomposition} (V_n , E_n ), n \geq 1.$ If $Ex(X_n)\neq 0$ and $V(X_n) << E^2(X_n),$ then
\begin{eqnarray*}
E(X_n=0)\rightarrow 0~\text{as}~n\rightarrow \infty
\end{eqnarray*}
\end{corollary}
\begin{proof}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Set $X := X_n$ and $t := |Ex(X_n)|$ in Chebyshev’s Inequality, and observe that $E (X_n = 0) \leq E (|X_n - Ex(X_n)| \geq |Ex(X_n)|)$ because $|X_n - Ex(X_n)| = |Ex(X_n)|$ when $X_n = 0.$
\end{proof}
\begin{theorem}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Let $G \in \mathcal{G}_{n,1/2}.$ For $0 \leq k \leq n,$ set $f(k) := \text{(n choose k)}2^{-\text{(k choose 2)}}$ and let $k^{*}$ be the least value of $k$ for which $f(k)$ is less than one. Then almost surely $\alpha(G)$ takes one of the three values $k^{*}-2,k^{*}-1,k^{*}.$
\end{theorem}
\begin{proof}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. As in the proof of related Theorem, the result is straightforward.
\end{proof}
\begin{corollary}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Let $G \in \mathcal{G}_{n,1/2}$ and let $f$ and $k^{*}$ be as defined in previous Theorem. Then either:
\begin{itemize}
\item[$(i).$] $f(k^{*}) << 1,$ in which case almost surely $\alpha(G)$ is equal to either $k^{*}-2$ or $k^{*}-1$, or
\item[$(ii).$] $f(k^{*}-1) >> 1,$ in which case almost surely $\alpha(G)$ is equal to either $k^{*}-1$ or $k^{*}.$
\end{itemize}
\end{corollary}
\begin{proof}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. The latter is straightforward.
\end{proof}
\begin{definition}(an Extreme Threshold).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Let $P$ be a monotone property of SuperHyperGraphs (one which is preserved when SuperHyperEdges are added). Then a \textbf{an Extreme Threshold} for $P$ is a function $f(n)$ such that:
\begin{itemize}
\item[$(i).$] if $p << f(n),$ then $G \in \mathcal{G}_{n,p}$ almost surely does not have $P,$
\item[$(ii).$] if $p >> f(n),$ then $G \in \mathcal{G}_{n,p}$ almost surely has $P.$
\end{itemize}
\end{definition}
\begin{definition}(an Extreme Balanced).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Let $F$ be a fixed an Extreme SuperHyperGraph. Then there is a threshold function for the property of containing a copy of $F$ as a an Extreme SubSuperHyperGraph is called \textbf{an Extreme Balanced}.
\end{definition}
\begin{theorem}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. Let $F$ be a nonempty balanced an Extreme SubSuperHyperGraph with $k$ SuperHyperVertices and $l$ SuperHyperEdges. Then $n^{-k/l}$ is a threshold function for the property of containing $F$ as a an Extreme SubSuperHyperGraph.
\end{theorem}
\begin{proof}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability {\tiny Eulerian-Type-Cycle-Decomposition}. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM1}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned an Extreme Figures in every an Extreme items.
\begin{itemize}
\item On the Figure \eqref{136NSHG1}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward. $E_1$ and $E_3$ are some empty an Extreme SuperHyperEdges but $E_2$ is a loop an Extreme SuperHyperEdge and $E_4$ is a an Extreme SuperHyperEdge. Thus in the terms of an Extreme SuperHyperNeighbor, there's only one an Extreme SuperHyperEdge, namely, $E_4.$ The an Extreme SuperHyperVertex, $V_3$ is an Extreme isolated means that there's no an Extreme SuperHyperEdge has it as a an Extreme endpoint. Thus the an Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_4\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_1,V_2,V_4\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=z^3.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG1.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG1}
\end{figure}
\item On the Figure \eqref{136NSHG2}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward. $E_1,E_2$ and $E_3$ are some empty an Extreme SuperHyperEdges but $E_4$ is a an Extreme SuperHyperEdge. Thus in the terms of an Extreme SuperHyperNeighbor, there's only one an Extreme SuperHyperEdge, namely, $E_4.$ The an Extreme SuperHyperVertex, $V_3$ is an Extreme isolated means that there's no an Extreme SuperHyperEdge has it as a an Extreme endpoint. Thus the an Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_4\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_1,V_2,V_4\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=z^3.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG2.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG2}
\end{figure}
\item On the Figure \eqref{136NSHG3}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_4\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_1,V_2,V_3\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=z^3.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG3.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG3}
\end{figure}
\item On the Figure \eqref{136NSHG4}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_4,E_1,E_5,E_4\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=z^4.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_1,V_2,\{N\},\{F\},V_3,\{H\},V_4,V_1\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=z^8.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG4.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG4}
\end{figure}
\item On the Figure \eqref{136NSHG5}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_1\},\{E_2\},\{E_3\},\{E_4\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=4z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_{i=1^5}\},\{V_{i=6^8},V_5\},\{V_{i=9^{12}},V_5\},\{V_{i={13}^{15}},V_5\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=2z^5+2z^4.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG5.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG5}
\end{figure}
\item On the Figure \eqref{136NSHG6}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_{i=1^{10},E_{i={23}^{32},E_{12},E_{22}\},\ldots\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=az^{22}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_i\in V_{NSHG}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=z^{|V_{NSHG}|}.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG6.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG6}
\end{figure}
\item On the Figure \eqref{136NSHG7}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_{i=1^{10},E_{17},E_{14},E_{13},E_{15}\},\ldots\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=z^{14}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_i\in V_{NSHG}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
= z^{|V_{NSHG}|}.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG7.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG7}
\end{figure}
\item On the Figure \eqref{136NSHG8}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_3\},\{E_2\},\{E_1\},\{E_4\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=4z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_{i=1^3},V_{12}\},\{V_{i=4^7},V_{13}\},\{V_{i=9^{12}},V_5\},\{V_{i=8^{11}},V_{14}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=2z^5+z^4+z^3.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG8.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG8}
\end{figure}
\item On the Figure \eqref{136NSHG9}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_{i=1^{10},E_{23},E_{12},E_{22}\},\ldots\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=az^{13}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_i\in V_{NSHG}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=z^{|V_{NSHG}|}.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG9.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG9}
\end{figure}
\item On the Figure \eqref{136NSHG10}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_1,E_6,E_2,E_7,E_3,E_4,E_5,E_1\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=z^{|E_{NSHG}|}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_i\in V_{NSHG}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
= z^{|V_{NSHG}|}.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG10.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG10}
\end{figure}
\item On the Figure \eqref{136NSHG11}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{E_1,E_6,E_5,E_4,E_7,E_1,\ldots\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=2z^3+4z^2+\ldots.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_i\in V_{NSHG}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
= z^{|V_{NSHG}|}.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG11.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG11}
\end{figure}
\item On the Figure \eqref{136NSHG12}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_1\},E_{i=2^6}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=z^5+5z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_1,V_2,V_3,V_7,V_8\},\ldots\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=z^5+\ldots.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG12.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG12}
\end{figure}
\item On the Figure \eqref{136NSHG13}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{E_1,E_6,E_5,E_4,E_2,E_3,E_7,E_{10},E_9,E_1,\ldots\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=2z^3+7z^2+\ldots.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_i\in V_{NSHG}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
= z^{|V_{NSHG}|}.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG13.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG13}
\end{figure}
\item On the Figure \eqref{136NSHG14}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_1\},\{E_2\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=2z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_1,V_3\},\{V_1,V_2\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=2z^2.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG14.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG14}
\end{figure}
\item On the Figure \eqref{136NSHG15}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_{i=1^5}\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=5z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_{\{E_{i=1^5}\}}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=5z^2.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG15.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG15}
\end{figure}
\item On the Figure \eqref{136NSHG16}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_{i=1^5}\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=\sum_{i=1}^5 z^{|\{E_{i=1^5}\}|}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_{\{E_{i=1^5}\}}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=\sum_{i=1}^5 z^{|\{V_{\{E_{i=1^5}\}}\}|}.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG16.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG16}
\end{figure}
\item On the Figure \eqref{136NSHG17}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_{i=1^6}\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=\sum_{i=1}^6 z^{|\{E_{i=1^6}\}|}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_{\{E_{i=1^6}\}}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=\sum_{i=1}^6 z^{|\{V_{\{E_{i=1^6}\}}\}|}.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG17.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG17}
\end{figure}
\item On the Figure \eqref{136NSHG18}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_{i=1^6}\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=\sum_{i=1}^6 z^{|\{E_{i=1^6}\}|}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_{\{E_{i=1^6}\}}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=\sum_{i=1}^6 z^{|\{V_{\{E_{i=1^6}\}}\}|}.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG18.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG18}
\end{figure}
\item On the Figure \eqref{136NSHG19}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_{i=1^{12}}\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=12z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_i\in V_{NSHG}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=z^{|V_{NSHG}|}.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG19.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG19}
\end{figure}
\item On the Figure \eqref{136NSHG20}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_{i=1^{10}}\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=\sum_{i=1}^{10} z^{|\{E_{i=1^{10}}\}|}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_{\{E_{i=1^{10}}\}}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=\sum_{i=1}^{10} z^{|\{V_{\{E_{i=1^{10}}\}}\}|}.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{136NSHG20.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{136NSHG20}
\end{figure}
\item On the Figure \eqref{95NHG1}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_2\},\{E_1\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=2z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_{E_2}\},\{V_1,V_3\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=z^{10}+z^2.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{95NHG1.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{95NHG1}
\end{figure}
\item On the Figure \eqref{95NHG2}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_{i=2^5}\},\ldots\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=\sum_{i=2}^5 z^{|\{E_{i=2^5}\}|}+\ldots.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_{\{E_{i=2^5}\}}\},\ldots\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=\sum_{i=2}^5 z^{|\{V_{\{E_{i=2^5}\}}\}|}+\ldots.
\end{eqnarray*}
\begin{figure}
\includegraphics[width=100mm]{95NHG2.png}
\caption{The an Extreme SuperHyperGraphs Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM1} }
\label{95NHG2}
\end{figure}
\end{itemize}
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperGraph $ESHG:(V,E).$ The all interior an Extreme SuperHyperVertices belong to any an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition} if for any of them, and any of other corresponded an Extreme SuperHyperVertex, some interior an Extreme SuperHyperVertices are mutually an Extreme SuperHyperNeighbors with no an Extreme exception at all minus all an Extreme SuperHypeNeighbors to any amount of them.
\end{proposition}
\begin{proposition}
Assume a connected non-obvious an Extreme SuperHyperGraph $ESHG:(V,E).$ There's only one an Extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only the maximum possibilities of the distinct interior an Extreme SuperHyperVertices inside of any given an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition} minus all an Extreme SuperHypeNeighbor to some of them but not all of them. In other words, there's only an unique an Extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only two distinct an Extreme SuperHyperVertices in an an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition}, minus all an Extreme SuperHypeNeighbor to some of them but not all of them.
\end{proposition}
\begin{proposition}
Assume a connected an Extreme SuperHyperGraph $ESHG:(V,E).$ If a an Extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$ an Extreme SuperHyperVertices, then the an Extreme cardinality of the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} is at least
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
It's straightforward that the an Extreme cardinality of the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} is at least the maximum an Extreme number of an Extreme SuperHyperVertices of the an Extreme SuperHyperEdges with the maximum number of the an Extreme SuperHyperEdges. In other words, the maximum number of the an Extreme SuperHyperEdges contains the maximum an Extreme number of an Extreme SuperHyperVertices are renamed to an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} in some cases but the maximum number of the an Extreme SuperHyperEdge with the maximum an Extreme number of an Extreme SuperHyperVertices, has the an Extreme SuperHyperVertices are contained in a an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{proposition}
\begin{proposition}
Assume a simple an Extreme SuperHyperGraph $ESHG:(V,E).$ Then the an Extreme number of type-result-R-{\tiny Eulerian-Type-Cycle-Decomposition} has, the least an Extreme cardinality, the lower sharp an Extreme bound for an Extreme cardinality, is the an Extreme cardinality of
$$V\setminus V\setminus\left\{a_{E},b_{E'},c_{E''},c_{E'''}\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
If there's a an Extreme type-result-R-{\tiny Eulerian-Type-Cycle-Decomposition} with the least an Extreme cardinality, the lower sharp an Extreme bound for cardinality.
\end{proposition}
\begin{proposition}
Assume a connected loopless an Extreme SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}
\\&&
=\{V_1,E_1,V_2,E_2,V_3,E_3,V_4,E_4,V_1\}.
\\&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}=z^4.
\\&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}=\{V_1,V_2,V_3,V_4,V_1\}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}=z^5.
\end{eqnarray*}
Is a an Extreme type-result-{\tiny Eulerian-Type-Cycle-Decomposition}. In other words, the least cardinality, the lower sharp bound for the cardinality, of a an Extreme type-result-{\tiny Eulerian-Type-Cycle-Decomposition} is the cardinality of
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}
\\&&
=\{V_1,E_1,V_2,E_2,V_3,E_3,V_4,E_4,V_1\}.
\\&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}=z^4.
\\&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}=\{V_1,V_2,V_3,V_4,V_1\}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}=z^5.
\end{eqnarray*}
\end{proposition}
\begin{proof}
Assume a connected loopless an Extreme SuperHyperGraph $ESHG:(V,E).$ The SuperHyperSet of the SuperHyperVertices $V\setminus V\setminus \{z\}$ isn't a quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition} since neither amount of an Extreme SuperHyperEdges nor amount of SuperHyperVertices where amount refers to the an Extreme number of SuperHyperVertices(-/SuperHyperEdges) more than one to form any kind of SuperHyperEdges or any number of SuperHyperEdges. Let us consider the an Extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
This an Extreme SuperHyperSet of the an Extreme SuperHyperVertices has the eligibilities to propose property such that there's no an Extreme SuperHyperVertex of a an Extreme SuperHyperEdge is common and there's an an Extreme SuperHyperEdge for all an Extreme SuperHyperVertices but the maximum an Extreme cardinality indicates that these an Extreme type-SuperHyperSets couldn't give us the an Extreme lower bound in the term of an Extreme sharpness. In other words, the an Extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
of the an Extreme SuperHyperVertices implies at least on-quasi-triangle style is up but sometimes the an Extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
of the an Extreme SuperHyperVertices is free-quasi-triangle and it doesn't make a contradiction to the supposition on the connected loopless an Extreme SuperHyperGraph $ESHG:(V,E).$ Thus the minimum case never happens in the generality of the connected loopless an Extreme SuperHyperGraphs. Thus if we assume in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is a quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition}. In other words, the least cardinality, the lower sharp bound for the cardinality, of a quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition} is the cardinality of
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Then we've lost some connected loopless an Extreme SuperHyperClasses of the connected loopless an Extreme SuperHyperGraphs titled free-triangle, on-triangle, and their quasi-types but the SuperHyperStable is only up in this quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition}. It's the contradiction to that fact on the generality. There are some counterexamples to deny this statement. One of them comes from the setting of the graph titled path and cycle as the counterexamples-classes or reversely direction star as the examples-classes, are well-known classes in that setting and they could be considered as the examples-classes and counterexamples-classes for the tight bound of
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Let $V\setminus V\setminus \{z\}$ in mind. There's no necessity on the SuperHyperEdge since we need at least two SuperHyperVertices to form a SuperHyperEdge. It doesn't withdraw the principles of the main definition since there's no condition to be satisfied but the condition is on the existence of the SuperHyperEdge instead of acting on the SuperHyperVertices. In other words, if there's a SuperHyperEdge, then the an Extreme SuperHyperSet has the necessary condition for the intended definition to be applied. Thus the $V\setminus V\setminus \{z\}$ is withdrawn not by the conditions of the main definition but by the necessity of the pre-condition on the usage of the main definition.
\\
The an Extreme structure of the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} decorates the an Extreme SuperHyperVertices don't have received any an Extreme connections so as this an Extreme style implies different versions of an Extreme SuperHyperEdges with the maximum an Extreme cardinality in the terms of an Extreme SuperHyperVertices are spotlight. The lower an Extreme bound is to have the maximum an Extreme groups of an Extreme SuperHyperVertices have perfect an Extreme connections inside each of SuperHyperEdges and the outside of this an Extreme SuperHyperSet doesn't matter but regarding the connectedness of the used an Extreme SuperHyperGraph arising from its an Extreme properties taken from the fact that it's simple. If there's no more than one an Extreme SuperHyperVertex in the targeted an Extreme SuperHyperSet, then there's no an Extreme connection. Furthermore, the an Extreme existence of one an Extreme SuperHyperVertex has no an Extreme effect to talk about the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}. Since at least two an Extreme SuperHyperVertices involve to make a title in the an Extreme background of the an Extreme SuperHyperGraph. The an Extreme SuperHyperGraph is obvious if it has no an Extreme SuperHyperEdge but at least two an Extreme SuperHyperVertices make the an Extreme version of an Extreme SuperHyperEdge. Thus in the an Extreme setting of non-obvious an Extreme SuperHyperGraph, there are at least one an Extreme SuperHyperEdge. It's necessary to mention that the word ``Simple'' is used as an Extreme adjective for the initial an Extreme SuperHyperGraph, induces there's no an Extreme appearance of the loop an Extreme version of the an Extreme SuperHyperEdge and this an Extreme SuperHyperGraph is said to be loopless. The an Extreme adjective ``loop'' on the basic an Extreme framework engages one an Extreme SuperHyperVertex but it never happens in this an Extreme setting. With these an Extreme bases, on a an Extreme SuperHyperGraph, there's at least one an Extreme SuperHyperEdge thus there's at least a an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} has the an Extreme cardinality of a an Extreme SuperHyperEdge. Thus, a an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} has the an Extreme cardinality at least a an Extreme SuperHyperEdge. Assume a an Extreme SuperHyperSet $V\setminus V\setminus \{z\}.$ This an Extreme SuperHyperSet isn't a an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} since either the an Extreme SuperHyperGraph is an obvious an Extreme SuperHyperModel thus it never happens since there's no an Extreme usage of this an Extreme framework and even more there's no an Extreme connection inside or the an Extreme SuperHyperGraph isn't obvious and as its consequences, there's a an Extreme contradiction with the term ``an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}'' since the maximum an Extreme cardinality never happens for this an Extreme style of the an Extreme SuperHyperSet and beyond that there's no an Extreme connection inside as mentioned in first an Extreme case in the forms of drawback for this selected an Extreme SuperHyperSet. Let
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Comes up. This an Extreme case implies having the an Extreme style of on-quasi-triangle an Extreme style on the every an Extreme elements of this an Extreme SuperHyperSet. Precisely, the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} is the an Extreme SuperHyperSet of the an Extreme SuperHyperVertices such that some an Extreme amount of the an Extreme SuperHyperVertices are on-quasi-triangle an Extreme style. The an Extreme cardinality of the v SuperHypeSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots,a_{E'},b_{E'},c_{E'},\ldots\right\}_{E,E'=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}$$
Is the maximum in comparison to the an Extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But the lower an Extreme bound is up. Thus the minimum an Extreme cardinality of the maximum an Extreme cardinality ends up the an Extreme discussion. The first an Extreme term refers to the an Extreme setting of the an Extreme SuperHyperGraph but this key point is enough since there's a an Extreme SuperHyperClass of a an Extreme SuperHyperGraph has no on-quasi-triangle an Extreme style amid some amount of its an Extreme SuperHyperVertices. This an Extreme setting of the an Extreme SuperHyperModel proposes a an Extreme SuperHyperSet has only some amount an Extreme SuperHyperVertices from one an Extreme SuperHyperEdge such that there's no an Extreme amount of an Extreme SuperHyperEdges more than one involving these some amount of these an Extreme SuperHyperVertices. The an Extreme cardinality of this an Extreme SuperHyperSet is the maximum and the an Extreme case is occurred in the minimum an Extreme situation. To sum them up, the an Extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Has the maximum an Extreme cardinality such that
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Contains some an Extreme SuperHyperVertices such that there's distinct-covers-order-amount an Extreme SuperHyperEdges for amount of an Extreme SuperHyperVertices taken from the an Extreme SuperHyperSet
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
It means that the an Extreme SuperHyperSet of the an Extreme SuperHyperVertices
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is a an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} for the an Extreme SuperHyperGraph as used an Extreme background in the an Extreme terms of worst an Extreme case and the common theme of the lower an Extreme bound occurred in the specific an Extreme SuperHyperClasses of the an Extreme SuperHyperGraphs which are an Extreme free-quasi-triangle.
\\
Assume a an Extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$ an Extreme number of the an Extreme SuperHyperVertices. Then every an Extreme SuperHyperVertex has at least no an Extreme SuperHyperEdge with others in common. Thus those an Extreme SuperHyperVertices have the eligibles to be contained in a an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}. Those an Extreme SuperHyperVertices are potentially included in a an Extreme style-R-{\tiny Eulerian-Type-Cycle-Decomposition}. Formally, consider
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
Are the an Extreme SuperHyperVertices of a an Extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z.$$ where the $\sim$ isn't an equivalence relation but only the symmetric relation on the an Extreme SuperHyperVertices of the an Extreme SuperHyperGraph. The formal definition is as follows. $$Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z$$ if and only if $Z_i$ and $Z_j$ are the an Extreme SuperHyperVertices and there's only and only one an Extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ between the an Extreme SuperHyperVertices $Z_i$ and $Z_j.$ The other definition for the an Extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ in the terms of an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} is
$$\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}.$$
This definition coincides with the definition of the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} but with slightly differences in the maximum an Extreme cardinality amid those an Extreme type-SuperHyperSets of the an Extreme SuperHyperVertices. Thus the an Extreme SuperHyperSet of the an Extreme SuperHyperVertices,
$$\max_z|\{Z_1,Z_2,\ldots,Z_z~|~Z_i\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}|_{\text{an Extreme cardinality}},$$
and
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
is formalized with mathematical literatures on the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}. Let $Z_i\overset E\sim Z_j,$ be defined as $Z_i$ and $Z_j$ are the an Extreme SuperHyperVertices belong to the an Extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}.$ Thus,
$$E=\{Z_1,Z_2,\ldots,Z_z~|~Z_i\overset E\sim Z_j,~i\neq j,~ i,j=1,2,\ldots,z\}.$$
Or
$$\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
But with the slightly differences,
\begin{eqnarray*}
&&
\text{an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}}=
\\&&
\{Z_1,Z_2,\ldots,Z_z~|~\forall i\neq j,~ i,j=1,2,\ldots,z,~\exists E_x,~Z_i\overset{E_x}\sim Z_j,\}.
\end{eqnarray*}
\begin{eqnarray*}
&&
\text{an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}}=
\\&&
V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.
\end{eqnarray*}
Thus $E\in E_{ESHG:(V,E)}$ is a an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition} where $E\in E_{ESHG:(V,E)}$ is fixed that means $E_x=E\in E_{ESHG:(V,E)}.$ for all an Extreme intended SuperHyperVertices but in a an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}, $E_x=E\in E_{ESHG:(V,E)}$ could be different and it's not unique. To sum them up, in a connected an Extreme SuperHyperGraph $ESHG:(V,E).$ If a an Extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has $z$ an Extreme SuperHyperVertices, then the an Extreme cardinality of the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} is at least
$$V\setminus (V\setminus\left\{a_{E},b_{E},c_{E},\ldots,z_{E}\right\}).$$
It's straightforward that the an Extreme cardinality of the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} is at least the maximum an Extreme number of an Extreme SuperHyperVertices of the an Extreme SuperHyperEdges with the maximum number of the an Extreme SuperHyperEdges. In other words, the maximum number of the an Extreme SuperHyperEdges contains the maximum an Extreme number of an Extreme SuperHyperVertices are renamed to an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} in some cases but the maximum number of the an Extreme SuperHyperEdge with the maximum an Extreme number of an Extreme SuperHyperVertices, has the an Extreme SuperHyperVertices are contained in a an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}.
\\
The obvious SuperHyperGraph has no an Extreme SuperHyperEdges. But the non-obvious an Extreme SuperHyperModel is up. The quasi-SuperHyperModel addresses some issues about the an Extreme optimal SuperHyperObject. It specially delivers some remarks on the an Extreme SuperHyperSet of the an Extreme SuperHyperVertices such that there's distinct amount of an Extreme SuperHyperEdges for distinct amount of an Extreme SuperHyperVertices up to all taken from that an Extreme SuperHyperSet of the an Extreme SuperHyperVertices but this an Extreme SuperHyperSet of the an Extreme SuperHyperVertices is either has the maximum an Extreme SuperHyperCardinality or it doesn't have maximum an Extreme SuperHyperCardinality. In a non-obvious SuperHyperModel, there's at least one an Extreme SuperHyperEdge containing at least all an Extreme SuperHyperVertices. Thus it forms a an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition} where the an Extreme completion of the an Extreme incidence is up in that. Thus it's, literarily, a an Extreme embedded R-{\tiny Eulerian-Type-Cycle-Decomposition}. The SuperHyperNotions of embedded SuperHyperSet and quasi-SuperHyperSet coincide. In the original setting, these types of SuperHyperSets only don't satisfy on the maximum SuperHyperCardinality. Thus the embedded setting is elected such that those SuperHyperSets have the maximum an Extreme SuperHyperCardinality and they're an Extreme SuperHyperOptimal. The less than two distinct types of an Extreme SuperHyperVertices are included in the minimum an Extreme style of the embedded an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}. The interior types of the an Extreme SuperHyperVertices are deciders. Since the an Extreme number of SuperHyperNeighbors are only affected by the interior an Extreme SuperHyperVertices. The common connections, more precise and more formal, the perfect unique connections inside the an Extreme SuperHyperSet for any distinct types of an Extreme SuperHyperVertices pose the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}. Thus an Extreme exterior SuperHyperVertices could be used only in one an Extreme SuperHyperEdge and in an Extreme SuperHyperRelation with the interior an Extreme SuperHyperVertices in that an Extreme SuperHyperEdge. In the embedded an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}, there's the usage of exterior an Extreme SuperHyperVertices since they've more connections inside more than outside. Thus the title ``exterior'' is more relevant than the title ``interior''. One an Extreme SuperHyperVertex has no connection, inside. Thus, the an Extreme SuperHyperSet of the an Extreme SuperHyperVertices with one SuperHyperElement has been ignored in the exploring to lead on the optimal case implying the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}. The an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} with the exclusion of the exclusion of all an Extreme SuperHyperVertices in one an Extreme SuperHyperEdge and with other terms, the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} with the inclusion of all an Extreme SuperHyperVertices in one an Extreme SuperHyperEdge, is a an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition}. To sum them up, in a connected non-obvious an Extreme SuperHyperGraph $ESHG:(V,E).$ There's only one an Extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only the maximum possibilities of the distinct interior an Extreme SuperHyperVertices inside of any given an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition} minus all an Extreme SuperHypeNeighbor to some of them but not all of them. In other words, there's only an unique an Extreme SuperHyperEdge $E\in E_{ESHG:(V,E)}$ has only two distinct an Extreme SuperHyperVertices in an an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition}, minus all an Extreme SuperHypeNeighbor to some of them but not all of them.
\\
The main definition of the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} has two titles. a an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition} and its corresponded quasi-maximum an Extreme R-SuperHyperCardinality are two titles in the terms of quasi-R-styles. For any an Extreme number, there's a an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition} with that quasi-maximum an Extreme SuperHyperCardinality in the terms of the embedded an Extreme SuperHyperGraph. If there's an embedded an Extreme SuperHyperGraph, then the an Extreme quasi-SuperHyperNotions lead us to take the collection of all the an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition}s for all an Extreme numbers less than its an Extreme corresponded maximum number. The essence of the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} ends up but this essence starts up in the terms of the an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition}, again and more in the operations of collecting all the an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition}s acted on the all possible used formations of the an Extreme SuperHyperGraph to achieve one an Extreme number. This an Extreme number is\\ considered as the equivalence class for all corresponded quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition}s. Let $z_{\text{an Extreme Number}},S_{\text{an Extreme SuperHyperSet}}$ and $G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}$ be a an Extreme number, a an Extreme SuperHyperSet and a an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}. Then
\begin{eqnarray*}
&&[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}=\{S_{\text{an Extreme SuperHyperSet}}~|
\\&&~S_{\text{an Extreme SuperHyperSet}}=G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}},
\\&&~|S_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&=z_{\text{an Extreme Number}}\}.
\end{eqnarray*}
As its consequences, the formal definition of the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} is re-formalized and redefined as follows.
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}\in\cup_{z_{\text{an Extreme Number}}}
\\&&
[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}=
\\&&
\cup_{z_{\text{an Extreme Number}}}
\\&&
\{S_{\text{an Extreme SuperHyperSet}}~|
\\&&~S_{\text{an Extreme SuperHyperSet}}=G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}},
\\&&~|S_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&=z_{\text{an Extreme Number}}\}.
\end{eqnarray*}
To get more precise perceptions, the follow-up expressions propose another formal technical definition for the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}.
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}=
\\&&
\{S\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}=
\\&&
\cup_{z_{\text{an Extreme Number}}}\{S_{\text{an Extreme SuperHyperSet}}~|
\\&&~S_{\text{an Extreme SuperHyperSet}}=G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}},
\\&&~|S_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&=z_{\text{an Extreme Number}}~|~
\\&&
|S_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&
=\max_{[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}}z_{\text{an Extreme Number}}
\}.
\end{eqnarray*}
In more concise and more convenient ways, the modified definition for the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} poses the upcoming expressions.
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}=
\\&&
\{S\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}~|~
\\&&
|S_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&
=\max_{[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}}z_{\text{an Extreme Number}}
\}.
\end{eqnarray*}
To translate the statement to this mathematical literature, the formulae will be revised.
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}=
\\&&
\{S\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}~|~
\\&&
|S_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&
=\max_{[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}}z_{\text{an Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
And then,
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}=
\\&&
\{S\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}~|~
\\&&
|S_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
To get more visions in the closer look-up, there's an overall overlook.
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}=
\\&&
\cup_{z_{\text{an Extreme Number}}}\{S_{\text{an Extreme SuperHyperSet}}~|
\\&&~S_{\text{an Extreme SuperHyperSet}}=G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}},
\\&&~|S_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}=
\\&&
\{S\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}=
\\&&
\cup_{z_{\text{an Extreme Number}}}\{S_{\text{an Extreme SuperHyperSet}}~|
\\&&~S_{\text{an Extreme SuperHyperSet}}=G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}},
\\&&~|S_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&=z_{\text{an Extreme Number}}~|~
\\&&
|S_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}=
\\&&
\{S\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}~|~
\\&&
|S_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&
=\max_{[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}}z_{\text{an Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}=
\\&&
\{S\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}~|~
\\&&
|S_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Now, the extension of these types of approaches is up. Since the new term, ``an Extreme SuperHyperNeighborhood'', could be redefined as the collection of the an Extreme SuperHyperVertices such that any amount of its an Extreme SuperHyperVertices are incident to a an Extreme SuperHyperEdge. It's, literarily, another name for ``an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}'' but, precisely, it's the generalization of ``an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}'' since ``an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}'' happens ``an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}'' in a an Extreme SuperHyperGraph as initial framework and background but ``an Extreme SuperHyperNeighborhood'' may not happens ``an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}'' in a an Extreme SuperHyperGraph as initial framework and preliminarily background since there are some ambiguities about the an Extreme SuperHyperCardinality arise from it. To get orderly keywords, the terms, ``an Extreme SuperHyperNeighborhood'', ``an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}'', and ``an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}'' are up.
\\
Thus, let \\$z_{\text{an Extreme Number}},N_{\text{an Extreme SuperHyperNeighborhood}}$ and \\$G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}$ be a an Extreme number, a an Extreme SuperHyperNeighborhood and a an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} and the new terms are up.
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}\in\cup_{z_{\text{an Extreme Number}}}
\\&&
[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}=
\\&&
\cup_{z_{\text{an Extreme Number}}}\{N_{\text{an Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{an Extreme SuperHyperNeighborhood}}|_{\text{an Extreme Cardinality}}
\\&&=\max_{[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}}z_{\text{an Extreme Number}}\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}=
\\&&
\{N_{\text{an Extreme SuperHyperNeighborhood}}
\\&&
\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}=
\\&&
\cup_{z_{\text{an Extreme Number}}}\{N_{\text{an Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&=z_{\text{an Extreme Number}}~|~
\\&&
|N_{\text{an Extreme SuperHyperNeighborhood}}|_{\text{an Extreme Cardinality}}
\\&&
=\max_{[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}}z_{\text{an Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}=
\\&&
\{N_{\text{an Extreme SuperHyperNeighborhood}}
\\&&
\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}~|~
\\&&
|N_{\text{an Extreme SuperHyperNeighborhood}}|_{\text{an Extreme Cardinality}}
\\&&
=\max_{[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}}z_{\text{an Extreme Number}}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}=
\\&&
\{N_{\text{an Extreme SuperHyperNeighborhood}}
\\&&
\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}~|~
\\&&
|N_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&
=\max_{[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}}z_{\text{an Extreme Number}}
\}.
\end{eqnarray*}
And with go back to initial structure,
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}=
\\&&
\cup_{z_{\text{an Extreme Number}}}\{N_{\text{an Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{an Extreme SuperHyperNeighborhood}}|_{\text{an Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}=
\\&&
\{N_{\text{an Extreme SuperHyperNeighborhood}}
\\&&
\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}=
\\&&
\cup_{z_{\text{an Extreme Number}}}\{N_{\text{an Extreme SuperHyperNeighborhood}}~|
\\&&~|N_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&=z_{\text{an Extreme Number}}~|~
\\&&
|N_{\text{an Extreme SuperHyperNeighborhood}}|_{\text{an Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}=
\\&&
\{N_{\text{an Extreme SuperHyperNeighborhood}}
\\&&
\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}~|~
\\&&
|N_{\text{an Extreme SuperHyperNeighborhood}}|_{\text{an Extreme Cardinality}}
\\&&
=\max_{[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}}z_{\text{an Extreme Number}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
\begin{eqnarray*}
&&G_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}=
\\&&
\{N_{\text{an Extreme SuperHyperNeighborhood}}
\\&&
\in\cup_{z_{\text{an Extreme Number}}}[z_{\text{an Extreme Number}}]_{\text{an Extreme Class}}~|~
\\&&
|N_{\text{an Extreme SuperHyperSet}}|_{\text{an Extreme Cardinality}}
\\&&
=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}
\}.
\end{eqnarray*}
Thus, in a connected an Extreme SuperHyperGraph $ESHG:(V,E).$ The all interior an Extreme SuperHyperVertices belong to any an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition} if for any of them, and any of other corresponded an Extreme SuperHyperVertex, some interior an Extreme SuperHyperVertices are mutually an Extreme SuperHyperNeighbors with no an Extreme exception at all minus all an Extreme SuperHypeNeighbors to any amount of them.
\\
To make sense with the precise words in the terms of ``R-', the follow-up illustrations are coming up.
\\
The following an Extreme SuperHyperSet of an Extreme SuperHyperVertices is the simple an Extreme type-SuperHyperSet of the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}.
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
The an Extreme SuperHyperSet of an Extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the simple an Extreme type-SuperHyperSet of the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}. The an Extreme SuperHyperSet of the an Extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is an \underline{\textbf{an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}}} $\mathcal{C}(ESHG)$ for an an Extreme SuperHyperGraph $ESHG:(V,E)$ is a an Extreme type-SuperHyperSet with \\\\\underline{\textbf{the maximum an Extreme cardinality}} of a an Extreme SuperHyperSet $S$ of an Extreme SuperHyperVertices such that there's no a an Extreme SuperHyperEdge amid some an Extreme SuperHyperVertices instead of all given by
\\
\underline{\textbf{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}} is related to the an Extreme SuperHyperSet of the an Extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
There's \underline{not} only \underline{\textbf{one}} an Extreme SuperHyperVertex \underline{\textbf{inside}} the intended an Extreme SuperHyperSet. Thus the non-obvious an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} is up. The obvious simple an Extreme type-SuperHyperSet called the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} is a an Extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{one}} an Extreme SuperHyperVertex. But the an Extreme SuperHyperSet of an Extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
doesn't have less than two SuperHyperVertices \underline{\textbf{inside}} the intended an Extreme SuperHyperSet since they've come from at least so far an SuperHyperEdge. Thus the non-obvious simple an Extreme type-SuperHyperSet of the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} \underline{\textbf{is}} up. To sum them up, the an Extreme SuperHyperSet of an Extreme SuperHyperVertices,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
\underline{\textbf{Is}} the non-obvious simple an Extreme type-SuperHyperSet of the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}. Since the an Extreme SuperHyperSet of the an Extreme SuperHyperVertices,
$$(V\setminus V\setminus \{x,z\})\cup\{xy\}$$ or $$(V\setminus V\setminus \{x,z\})\cup\{zy\}$$
is an an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} $\mathcal{C}(ESHG)$ for an an Extreme SuperHyperGraph $ESHG:(V,E)$ is the an Extreme SuperHyperSet $S$ of an Extreme SuperHyperVertices such that there's no a an Extreme SuperHyperEdge for some an Extreme SuperHyperVertices instead of all given by that an Extreme type-SuperHyperSet called the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} \underline{\textbf{and}} it's an an Extreme \underline{\textbf{ {\tiny Eulerian-Type-Cycle-Decomposition}}}. Since it\underline{\textbf{'s}} \\\\\underline{\textbf{the maximum an Extreme cardinality}} of a an Extreme SuperHyperSet $S$ of an Extreme SuperHyperVertices such that there's no a an Extreme SuperHyperEdge for some amount an Extreme SuperHyperVertices instead of all given by that an Extreme type-SuperHyperSet called the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}. There isn't only less than two an Extreme SuperHyperVertices \underline{\textbf{inside}} the intended an Extreme SuperHyperSet,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Thus the non-obvious an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition},
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
is up. The non-obvious simple an Extreme type-SuperHyperSet of the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
Is the an Extreme SuperHyperSet, not:
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
does includes only less than two SuperHyperVertices in a connected an Extreme SuperHyperGraph $ESHG:(V,E)$ but it's impossible in the case, they've corresponded to an SuperHyperEdge. It's interesting to mention that the only non-obvious simple an Extreme type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}''}}
\end{center}
amid those obvious[non-obvious] simple an Extreme type-SuperHyperSets called the
\begin{center}
\underline{\textbf{an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}}},
\end{center}
is only and only
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected an Extreme SuperHyperGraph $ESHG:(V,E)$ with a illustrated SuperHyperModeling. It's also, not only a an Extreme free-triangle embedded SuperHyperModel and a an Extreme on-triangle embedded SuperHyperModel but also it's a an Extreme stable embedded SuperHyperModel. But all only non-obvious simple an Extreme type-SuperHyperSets of the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} amid those obvious simple an Extreme type-SuperHyperSets of the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}, are
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
In a connected an Extreme SuperHyperGraph $ESHG:(V,E).$
\\
To sum them up, assume a connected loopless an Extreme SuperHyperGraph $ESHG:(V,E).$ Then in the worst case, literally,
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
is a an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}. In other words, the least cardinality, the lower sharp bound for the cardinality, of a an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} is the cardinality of
$$V\setminus V\setminus\left\{a_{E},b_{E},c_{E},\ldots\right\}_{E=\left\{E\in E_{ESHG:(V,E)}~|~|E|=\max\left\{|E|~|~E\in E_{ESHG:(V,E)}\right\}\right\}}.$$
\\
To sum them up, in a connected an Extreme SuperHyperGraph $ESHG:(V,E).$ The all interior an Extreme SuperHyperVertices belong to any an Extreme quasi-R-{\tiny Eulerian-Type-Cycle-Decomposition} if for any of them, and any of other corresponded an Extreme SuperHyperVertex, some interior an Extreme SuperHyperVertices are mutually an Extreme SuperHyperNeighbors with no an Extreme exception at all minus all an Extreme SuperHypeNeighbors to any amount of them.
\\
Assume a connected an Extreme SuperHyperGraph $ESHG:(V,E).$ Let a an Extreme SuperHyperEdge $ESHE: E\in E_{ESHG:(V,E)}$ has some an Extreme SuperHyperVertices $r.$ Consider all an Extreme numbers of those an Extreme SuperHyperVertices from that an Extreme SuperHyperEdge excluding excluding more than $r$ distinct an Extreme SuperHyperVertices, exclude to any given an Extreme SuperHyperSet of the an Extreme SuperHyperVertices. Consider there's a an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} with the least cardinality, the lower sharp an Extreme bound for an Extreme cardinality. Assume a connected an Extreme SuperHyperGraph $ESHG:(V,E).$ The an Extreme SuperHyperSet of the an Extreme SuperHyperVertices $ V_{ESHE}\setminus \{z\}$ is a an Extreme SuperHyperSet $S$ of the an Extreme SuperHyperVertices such that there's a an Extreme SuperHyperEdge to have some an Extreme SuperHyperVertices uniquely but it isn't a an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}. Since it doesn't have \\\\\underline{\textbf{the maximum an Extreme cardinality}} of a an Extreme SuperHyperSet $S$ of an Extreme SuperHyperVertices such that there's a an Extreme SuperHyperEdge to have some SuperHyperVertices uniquely. The an Extreme SuperHyperSet of the an Extreme SuperHyperVertices $V_{ESHE}\cup \{z\}$ is the maximum an Extreme cardinality of a an Extreme SuperHyperSet $S$ of an Extreme SuperHyperVertices but it isn't a an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}. Since it \textbf{\underline{doesn't do}} the an Extreme procedure such that such that there's a an Extreme SuperHyperEdge to have some an Extreme SuperHyperVertices uniquely [there are at least one an Extreme SuperHyperVertex outside implying there's, sometimes in the connected an Extreme SuperHyperGraph $ESHG:(V,E),$ a an Extreme SuperHyperVertex, titled its an Extreme SuperHyperNeighbor, to that an Extreme SuperHyperVertex in the an Extreme SuperHyperSet $S$ so as $S$ doesn't do ``the an Extreme procedure''.]. There's only \textbf{\underline{one}} an Extreme SuperHyperVertex \textbf{\underline{outside}} the intended an Extreme SuperHyperSet, $V_{ESHE}\cup \{z\},$ in the terms of an Extreme SuperHyperNeighborhood. Thus the obvious an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}, $V_{ESHE}$ is up. The obvious simple an Extreme type-SuperHyperSet of the an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition}, $V_{ESHE},$ \textbf{\underline{is}} a an Extreme SuperHyperSet, $V_{ESHE},$ \textbf{\underline{includes}} only \textbf{\underline{all}} an Extreme SuperHyperVertices does forms any kind of an Extreme pairs are titled \underline{an Extreme SuperHyperNeighbors} in a connected an Extreme SuperHyperGraph $ESHG:(V,E).$ Since the an Extreme SuperHyperSet of the an Extreme SuperHyperVertices $V_{ESHE},$ is the \textbf{\underline{maximum an Extreme SuperHyperCardinality}} of a an Extreme SuperHyperSet $S$ of an Extreme SuperHyperVertices \textbf{\underline{such that}} there's a an Extreme SuperHyperEdge to have some an Extreme SuperHyperVertices uniquely. Thus, in a connected an Extreme SuperHyperGraph $ESHG:(V,E).$ Any an Extreme R-{\tiny Eulerian-Type-Cycle-Decomposition} only contains all interior an Extreme SuperHyperVertices and all exterior an Extreme SuperHyperVertices from the unique an Extreme SuperHyperEdge where there's any of them has all possible an Extreme SuperHyperNeighbors in and there's all an Extreme SuperHyperNeighborhoods in with no exception minus all an Extreme SuperHypeNeighbors to some of them not all of them but everything is possible about an Extreme SuperHyperNeighborhoods and an Extreme SuperHyperNeighbors out.
\\
The SuperHyperNotion, namely, {\tiny Eulerian-Type-Cycle-Decomposition}, is up. There's neither empty SuperHyperEdge nor loop SuperHyperEdge. The following an Extreme SuperHyperSet of an Extreme SuperHyperEdges[SuperHyperVertices] is the simple an Extreme type-SuperHyperSet of the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}.
The an Extreme SuperHyperSet of an Extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}
\\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}
\\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial} \\&&=az^{s}+bz^{t}.
\end{eqnarray*}
is the simple an Extreme type-SuperHyperSet of the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}. The an Extreme SuperHyperSet of the an Extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}
\\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}
\\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial} \\&&=az^{s}+bz^{t}.
\end{eqnarray*}
Is an \underline{\textbf{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}} $\mathcal{C}(ESHG)$ for an an Extreme SuperHyperGraph $ESHG:(V,E)$ is a an Extreme type-SuperHyperSet with \\\\\underline{\textbf{the maximum an Extreme cardinality}} of a an Extreme SuperHyperSet $S$ of an Extreme SuperHyperEdges[SuperHyperVertices] such that there's no an Extreme SuperHyperVertex of a an Extreme SuperHyperEdge is common and there's an an Extreme SuperHyperEdge for all an Extreme SuperHyperVertices. There are \underline{not} only \underline{\textbf{two}} an Extreme SuperHyperVertices \underline{\textbf{inside}} the intended an Extreme SuperHyperSet. Thus the non-obvious an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} is up. The obvious simple an Extreme type-SuperHyperSet called the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} is a an Extreme SuperHyperSet \underline{\textbf{includes}} only \underline{\textbf{two}} an Extreme SuperHyperVertices. But the an Extreme SuperHyperSet of the an Extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}
\\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}
\\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}
\\&&
=az^{s}+bz^{t}.
\end{eqnarray*}
Doesn't have less than three SuperHyperVertices \underline{\textbf{inside}} the intended an Extreme SuperHyperSet. Thus the non-obvious simple an Extreme type-SuperHyperSet of the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} \underline{\textbf{is}} up. To sum them up, the an Extreme SuperHyperSet of the an Extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}
\\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}
\\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial} \\&&=az^{s}+bz^{t}.
\end{eqnarray*}\underline{\textbf{Is}} the non-obvious simple an Extreme type-SuperHyperSet of the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}. Since the an Extreme SuperHyperSet of the an Extreme SuperHyperEdges[SuperHyperVertices],
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}
\\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}
\\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial} \\&&=az^{s}+bz^{t}.
\end{eqnarray*}
Is an an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} $\mathcal{C}(ESHG)$ for an an Extreme SuperHyperGraph $ESHG:(V,E)$ is the an Extreme SuperHyperSet $S$ of an Extreme SuperHyperVertices such that there's no a an Extreme SuperHyperEdge for some an Extreme SuperHyperVertices given by that an Extreme type-SuperHyperSet called the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} \underline{\textbf{and}} it's an an Extreme \underline{\textbf{ {\tiny Eulerian-Type-Cycle-Decomposition}}}. Since it\underline{\textbf{'s}} \\\\\underline{\textbf{the maximum an Extreme cardinality}} of a an Extreme SuperHyperSet $S$ of an Extreme SuperHyperEdges[SuperHyperVertices] such that there's no an Extreme SuperHyperVertex of a an Extreme SuperHyperEdge is common and there's an an Extreme SuperHyperEdge for all an Extreme SuperHyperVertices. There aren't only less than three an Extreme SuperHyperVertices \underline{\textbf{inside}} the intended an Extreme SuperHyperSet,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}
\\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}
\\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial} \\&&=az^{s}+bz^{t}.
\end{eqnarray*}
Thus the non-obvious an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}, \begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}
\\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}
\\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial} \\&&=az^{s}+bz^{t}.
\end{eqnarray*}
Is up. The obvious simple an Extreme type-SuperHyperSet of the an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}, not: \begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}
\\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}
\\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial} \\&&=az^{s}+bz^{t}.
\end{eqnarray*}
Is the an Extreme SuperHyperSet, not:
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}
\\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}
\\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial} \\&&=az^{s}+bz^{t}.
\end{eqnarray*}
Does includes only less than three SuperHyperVertices in a connected an Extreme SuperHyperGraph $ESHG:(V,E).$ It's interesting to mention that the only non-obvious simple an Extreme type-SuperHyperSet called the
\begin{center}
\underline{\textbf{``an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}''}}
\end{center}
amid those obvious[non-obvious] simple an Extreme type-SuperHyperSets called the
\begin{center}
\underline{\textbf{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}},
\end{center}
is only and only
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}
\\&&=\{E_{2i-1}\}_{i=1}^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}
\\&&=2z^{\lfloor\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{2}\rfloor}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition}}=\{V_i\}_{i=1}^{s},\{V_j\}_{j=1}^{t}.
\\&&
\mathcal{C}(NSHG)_{an Extreme R-Quasi-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial} \\&&=az^{s}+bz^{t}.
\end{eqnarray*}
In a connected an Extreme SuperHyperGraph $ESHG:(V,E).$
\end{proof}
\section{The an Extreme Departures on The Theoretical Results Toward Theoretical Motivations}
The previous an Extreme approach apply on the upcoming an Extreme results on an Extreme SuperHyperClasses.
\begin{proposition}
Assume a connected an Extreme SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_{i=1^{|E_{NSHG}|}\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=\sum_{i=1}^{|E_{NSHG}}|} z^{|\{E_{i=1^{|E_{NSHG}|}\}|}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_{\{E_{i=1^{|E_{NSHG}}|}\}}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=\sum_{i=1}^{|E_{NSHG}}|} z^{|\{V_{\{E_{i=1^{|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 an 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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM18a}
In the Figure \eqref{136NSHG18a}, the connected an Extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The an Extreme SuperHyperSet, in the an Extreme SuperHyperModel \eqref{136NSHG18a}, is the SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\begin{figure}
\includegraphics[width=100mm]{136NSHG18.png}
\caption{a an Extreme SuperHyperPath Associated to the Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the Example \eqref{136EXM18a}}
\label{136NSHG18a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_i\}_{i=1^{|E_{NSHG}|}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=z^{|E_{NSHG}|}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_i\}_{i=1^{|V_{NSHG}|}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=z^{|V_{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 an 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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM19a}
In the Figure \eqref{136NSHG19a}, the connected an Extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained an Extreme SuperHyperSet, in the an Extreme SuperHyperModel \eqref{136NSHG19a},
is the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\begin{figure}
\includegraphics[width=100mm]{136NSHG19.png}
\caption{a an Extreme SuperHyperCycle Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM19a}}
\label{136NSHG19a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperStar $ESHS:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_{i=1^{|E_{NSHG}|}\}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=\sum_{i=1}^{|E_{NSHG}|} z^{|\{E_{i=1^{|E_{NSHG}|}\}|}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_{\{E_{i=1^{|E_{NSHG}|}\}}\}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=\sum_{i=1}^{|E_{NSHG}|} z^{|\{V_{\{E_{i=1^{|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*}
be a longest path taken a connected an 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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM20a}
In the Figure \eqref{136NSHG20a}, the connected an Extreme SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperStar $ESHS:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG20a}, is the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\begin{figure}
\includegraphics[width=100mm]{136NSHG20.png}
\caption{a an Extreme SuperHyperStar Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an Extreme Example \eqref{136EXM20a}}
\label{136NSHG20a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_i,E_j\}_{E_i\cap E_j \neq \emptyset}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=z^{|\{E_i,E_j\}_{E_i\cap E_j \neq \emptyset}|}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_i,V_j\}_{V_i\in P^r_{NSHG},V_j\in P^s_{NSHG},r\neq s}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=z^{\{V_i,V_j\}_{V_i\in P^r_{NSHG},V_j\in P^s_{NSHG},r\neq s}}.
\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 an 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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. The latter is straightforward. Then there's no at least one SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} taken from a connected an Extreme SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-an 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 an Extreme Figure \eqref{136NSHG21a}, the connected an Extreme SuperHyperBipartite $ESHB:(V,E),$ is an Extreme highlighted and an Extreme featured. The obtained an Extreme SuperHyperSet, by the an Extreme Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperBipartite $ESHB:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG21a}, is the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\begin{figure}
\includegraphics[width=100mm]{136NSHG21.png}
\caption{an Extreme SuperHyperBipartite an Extreme Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the Example \eqref{136EXM21a}}
\label{136NSHG21a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_i,E_j\}_{E_i\cap E_j \neq \emptyset}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=z^{|\{E_i,E_j\}_{E_i\cap E_j \neq \emptyset}|}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_i,V_j\}_{V_i\in P^r_{NSHG},V_j\in P^s_{NSHG},r\neq s}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=z^{\{V_i,V_j\}_{V_i\in P^r_{NSHG},V_j\in P^s_{NSHG},r\neq s}}.
\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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} taken from a connected an 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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. The latter is straightforward. Then there's no at least one SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyper{\tiny Eulerian-Type-Cycle-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 an Extreme SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-an 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 an Extreme SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, the connected an Extreme SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and an Extreme featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperMultipartite $ESHM:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG22a}, is the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\begin{figure}
\includegraphics[width=100mm]{136NSHG22.png}
\caption{a an Extreme SuperHyperMultipartite Associated to the Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the Example \eqref{136EXM22a}}
\label{136NSHG22a}
\end{figure}
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperWheel $ESHW:(V,E\cup E^{*}).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{E_i\}_{i=1^{|E_{NSHG}|+1}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme {\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}
\\&&
=z^{|E_{NSHG}|+1}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition}}}
\\&&
=\{\{V_i\}_{i=1^{|V_{NSHG}|}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme V-{\tiny Eulerian-Type-Cycle-Decomposition} SuperHyperPolynomial}}}
\\&&
=z^{|V_{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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} taken from a connected an 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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. The latter is straightforward. Then there's at least one SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} could be applied. The unique embedded SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} proposes some longest SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} excerpt from some representatives. The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM23a}
In the an Extreme Figure \eqref{136NSHG23a}, the connected an Extreme SuperHyperWheel $NSHW:(V,E),$ is an Extreme highlighted and featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperWheel $ESHW:(V,E),$
in the an Extreme SuperHyperModel \eqref{136NSHG23a}, is the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\begin{figure}
\includegraphics[width=100mm]{136NSHG23.png}
\caption{a an Extreme SuperHyperWheel an Extreme Associated to the an Extreme Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in the an 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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, and the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, some general results are introduced.
\begin{remark}
Let remind that the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} is ``redefined'' on the positions of the alphabets.
\end{remark}
\begin{corollary}
Assume an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. Then
\begin{eqnarray*}
&& an Extreme ~SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}=\\&&\{the SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} of the SuperHyperVertices ~|~\\&&\max|SuperHyperOffensive \\&&SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}
\\&&
|_{an Extreme cardinality amid those SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.}\}
\end{eqnarray*}
plus one an 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 an Extreme SuperHyperGraph on the same identical letter of the alphabet. Then the notion of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} and SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} coincide.
\end{corollary}
\begin{corollary}
Assume a an Extreme SuperHyperGraph on the same identical letter of the alphabet. Then a conseNeighborive sequence of the SuperHyperVertices is a an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} if and only if it's a SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{corollary}
\begin{corollary}
Assume a an Extreme SuperHyperGraph on the same identical letter of the alphabet. Then a conseNeighborive sequence of the SuperHyperVertices is a strongest SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} if and only if it's a longest SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{corollary}
\begin{corollary}
Assume SuperHyperClasses of a an Extreme SuperHyperGraph on the same identical letter of the alphabet. Then its an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} is its SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} and reversely.
\end{corollary}
\begin{corollary}
Assume a an Extreme SuperHyperPath(-/SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel) on the same identical letter of the alphabet. Then its an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} is its SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} and reversely.
\end{corollary}
\begin{corollary}
Assume a an Extreme SuperHyperGraph. Then its an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} isn't well-defined if and only if its SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} isn't well-defined.
\end{corollary}
\begin{corollary}
Assume SuperHyperClasses of a an Extreme SuperHyperGraph. Then its an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} isn't well-defined if and only if its SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} isn't well-defined.
\end{corollary}
\begin{corollary}
Assume a an Extreme SuperHyperPath(-/SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). Then its an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} isn't well-defined if and only if its SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} isn't well-defined.
\end{corollary}
\begin{corollary}
Assume a an Extreme SuperHyperGraph. Then its an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} is well-defined if and only if its SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} is well-defined.
\end{corollary}
\begin{corollary}
Assume SuperHyperClasses of a an Extreme SuperHyperGraph. Then its an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} is well-defined if and only if its SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} is well-defined.
\end{corollary}
\begin{corollary}
Assume a an Extreme SuperHyperPath(-/SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). Then its an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} is well-defined if and only if its SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} is well-defined.
\end{corollary}
%
\begin{proposition}
Let $ESHG:(V,E)$ be a an Extreme SuperHyperGraph. Then $V$ is
\begin{itemize}
\item[$(i):$] the dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] the strong dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] the connected dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] the $\delta$-dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] the strong $\delta$-dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] the connected $\delta$-dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $NTG:(V,E,\sigma,\mu)$ be a an Extreme SuperHyperGraph. Then $\emptyset$ is
\begin{itemize}
\item[$(i):$] the SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] the strong SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] the connected defensive SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] the $\delta$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] the strong $\delta$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] the connected $\delta$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a an Extreme SuperHyperGraph. Then an independent SuperHyperSet is
\begin{itemize}
\item[$(i):$] the SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] the strong SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] the connected SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] the $\delta$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] the strong $\delta$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] the connected $\delta$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a an Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}/SuperHyperPath. Then $V$ is a maximal
\begin{itemize}
\item[$(i):$] SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] strong SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] connected SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] strong $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] connected $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\end{itemize}
Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a an Extreme SuperHyperGraph which is a SuperHyperUniform SuperHyperWheel. Then $V$ is a maximal
\begin{itemize}
\item[$(i):$] dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] strong dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] connected dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] strong $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] connected $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\end{itemize}
Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide.
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a an Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}/SuperHyperPath. Then the number of
\begin{itemize}
\item[$(i):$] the SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] the SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] the connected SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] the $\mathcal{O}(ESHG)$-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] the strong $\mathcal{O}(ESHG)$-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] the connected $\mathcal{O}(ESHG)$-SuperHyper{\tiny Eulerian-Type-Cycle-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 an Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyperWheel. Then the number of
\begin{itemize}
\item[$(i):$] the dual SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] the dual SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] the dual connected SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] the dual $\mathcal{O}(ESHG)$-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] the strong dual $\mathcal{O}(ESHG)$-SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] the connected dual $\mathcal{O}(ESHG)$-SuperHyper{\tiny Eulerian-Type-Cycle-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 an 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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] strong dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] connected dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] strong $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] connected $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a an 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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] strong SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] connected SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] $\delta$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] strong $\delta$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] connected $\delta$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a an Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyperStar/SuperHyperComplete SuperHyperBipartite/SuperHyperComplete SuperHyperMultipartite. Then Then the number of \begin{itemize}
\item[$(i):$] dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] strong dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] connected dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] strong $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] connected $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-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 an Extreme SuperHyperGraph. The number of connected component is $|V-S|$ if there's a SuperHyperSet which is a dual
\begin{itemize}
\item[$(i):$] SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] strong SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] connected SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] strong 1-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] connected 1-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a an Extreme SuperHyperGraph. Then the number is at most $\mathcal{O}(ESHG)$ and the an Extreme number is at most $\mathcal{O}_n(ESHG).$
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a an Extreme SuperHyperGraph which is SuperHyperComplete. The number is $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and the an 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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] strong
SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] connected SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] strong $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] connected $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a an Extreme SuperHyperGraph which is $\emptyset.$ The number is $0$ and the an Extreme number is $0,$ for an independent SuperHyperSet in the setting of dual
\begin{itemize}
\item[$(i):$] SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] strong SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] connected SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] $0$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] strong $0$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] connected $0$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a an Extreme SuperHyperGraph which is SuperHyperComplete. Then there's no independent SuperHyperSet.
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a an Extreme SuperHyperGraph which is SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}/SuperHyperPath/SuperHyperWheel. The number is $\mathcal{O}(ESHG:(V,E))$ and the an Extreme number is $\mathcal{O}_n(ESHG:(V,E)),$ in the setting of a dual
\begin{itemize}
\item[$(i):$] SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] strong SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] connected SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] strong $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] connected $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a an Extreme SuperHyperGraph which is SuperHyperStar/complete SuperHyperBipartite/complete SuperHyperMultiPartite. The number is $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and the an 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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii):$] strong SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii):$] connected SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iv):$] $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(v):$] strong $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(vi):$] connected $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $\mathcal{NSHF}:(V,E)$ be a SuperHyperFamily of the $ESHGs:(V,E)$ an 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 an Extreme SuperHyperGraphs.
\end{proposition}
%
\begin{proposition}
Let $ESHG:(V,E)$ be a strong an Extreme SuperHyperGraph. If $S$ is a dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-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 an Extreme SuperHyperGraph. If $S$ is a dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, then
\begin{itemize}
\item[$(i)$]
$S$ is SuperHyper{\tiny Eulerian-Type-Cycle-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 an 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 an 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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an even SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. Then
\begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n}\}$ is a dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be an odd SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. Then
\begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n-1}\}$ is a dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $\mathcal{NSHF}:(V,E)$ be a $m$-SuperHyperFamily of an Extreme SuperHyperStars with common an Extreme SuperHyperVertex SuperHyperSet. Then
\begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{c_1,c_2,\cdots,c_m\}$ is a dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 an 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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 an Extreme SuperHyperVertex SuperHyperSet. Then
\begin{itemize}
\item[$(i)$]
the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ is a dual SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} for $\mathcal{NSHF}:(V,E).$
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a strong an Extreme SuperHyperGraph. Then following statements hold;
\begin{itemize}
\item[$(i)$] if $s\geq t$ and a SuperHyperSet $S$ of SuperHyperVertices is an t-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, then $S$ is an s-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii)$] if $s\leq t$ and a SuperHyperSet $S$ of SuperHyperVertices is a dual t-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, then $S$ is a dual s-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a strong an 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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, then $S$ is an s-SuperHyperPowerful SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii)$] if $s\leq t$ and a SuperHyperSet $S$ of SuperHyperVertices is a dual t-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, then $S$ is a dual s-SuperHyperPowerful SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ be a[an] [V-]SuperHyperUniform-strong-an 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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is an V-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-an 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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an V-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-an 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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-an 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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-an Extreme SuperHyperGraph which is SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(ii)$] $\forall a\in V\setminus S,~|N_s(a)\cap S|> 2$ if $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an 2-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\begin{proposition}
Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-an Extreme SuperHyperGraph which is SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition};
\item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is an 2-SuperHyperDefensive SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\end{itemize}
\end{proposition}
\section{an Extreme Applications in Cancer's an Extreme Recognition}
The cancer is the an Extreme disease but the an Extreme model is going to figure out what's going on this an Extreme phenomenon. The special an Extreme case of this an 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 an Extreme recognition of the cancer could help to find some an Extreme treatments for this an Extreme disease.
\\
In the following, some an Extreme steps are an Extreme devised on this disease.
\begin{description}
\item[Step 1. (an Extreme Definition)] The an Extreme recognition of the cancer in the long-term an Extreme function.
\item[Step 2. (an Extreme Issue)] The specific region has been assigned by the an Extreme model [it's called an Extreme SuperHyperGraph] and the long an 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 an Extreme SuperHyperGraph] to have convenient perception on what's happened and what's done.
\item[Step 3. (an Extreme Model)]
There are some specific an Extreme models, which are well-known and they've got the names, and some general an Extreme models. The moves and the an Extreme traces of the cancer on the complex tracks and between complicated groups of cells could be fantasized by a an Extreme SuperHyperPath(-/SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). The aim is to find either the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} or the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} in those an Extreme an Extreme SuperHyperModels.
\section{Case 1: The Initial an Extreme Steps Toward an Extreme SuperHyperBipartite as an Extreme SuperHyperModel}
\item[Step 4. (an Extreme Solution)]
In the an Extreme Figure \eqref{136NSHGaa21aa}, the an Extreme SuperHyperBipartite is an Extreme highlighted and an Extreme featured.
\begin{figure}
\includegraphics[width=100mm]{136NSHG21.png}
\caption{a an Extreme SuperHyperBipartite Associated to the Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}}
\label{136NSHGaa21aa}
\end{figure}
\\ By using the an Extreme Figure \eqref{136NSHGaa21aa} and the Table \eqref{136TBLaa21aa}, the an Extreme SuperHyperBipartite is obtained.
\\
The obtained an Extreme SuperHyperSet, by the an Extreme Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperBipartite $ESHB:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHGaa21aa},
is the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The an 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 an Extreme Steps Toward an Extreme SuperHyperMultipartite as an Extreme SuperHyperModel}
\item[Step 4. (an Extreme Solution)]
In the an Extreme Figure \eqref{136NSHGaa22aa}, the an Extreme SuperHyperMultipartite is an Extreme highlighted and an Extreme featured.
\begin{figure}
\includegraphics[width=100mm]{136NSHG22.png}
\caption{a an Extreme SuperHyperMultipartite Associated to the Notions of an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}}
\label{136NSHGaa22aa}
\end{figure}
\\
By using the an Extreme Figure \eqref{136NSHGaa22aa} and the Table \eqref{136TBLaa22aa}, the an Extreme SuperHyperMultipartite is obtained.
\\
The obtained an Extreme SuperHyperSet, by the an Extreme Algorithm in previous result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperMultipartite $ESHM:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHGaa22aa}, is the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}.
\begin{table}
\centering
\caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The an 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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} and the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} and the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-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 SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} and the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}?
\end{question}
\begin{problem}
The SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} and the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} do a SuperHyperModel for the Cancer's recognitions and they're based on SuperHyper{\tiny Eulerian-Type-Cycle-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 an Extreme SuperHyperGraphs more understandable. In this endeavor, two SuperHyperNotions are defined on the SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}. For that sake in the second definition, the main definition of the an Extreme SuperHyperGraph is redefined on the position of the alphabets. Based on the new definition for the an Extreme SuperHyperGraph, the new SuperHyperNotion, an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, finds the convenient background to implement some results based on that. Some SuperHyperClasses and some an 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, SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}, the new SuperHyperClasses and SuperHyperClasses, are introduced. Some general results are gathered in the section on the SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition} and the an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-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 an 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 `` SuperHyper{\tiny Eulerian-Type-Cycle-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 an Extreme SuperHyperGraph} &\textcolor{black}{1. }\textcolor{blue}{General Results} \\ &
\\
\textcolor{black}{2. }\textcolor{red}{ SuperHyper{\tiny Eulerian-Type-Cycle-Decomposition}}& \\ &
\\
\textcolor{black}{3. } \textcolor{red}{an Extreme SuperHyper{\tiny Eulerian-Type-Cycle-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{
an Extreme SuperHyperDuality But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Different an Extreme Types of an Extreme SuperHyperDuality).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a an 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{an 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{an 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{an 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{an 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{an Extreme SuperHyperDuality} if it's either of an Extreme e-SuperHyperDuality, an Extreme re-SuperHyperDuality, an Extreme v-SuperHyperDuality, and an Extreme rv-SuperHyperDuality.
\end{itemize}
\end{definition}
\begin{definition}((an Extreme) SuperHyperDuality).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a an 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 an Extreme e-SuperHyperDuality, an Extreme re-SuperHyperDuality, an Extreme v-SuperHyperDuality, and an 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 conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality;
\item[$(ii)$]
a \textbf{an Extreme SuperHyperDuality} if it's either of an Extreme e-SuperHyperDuality, an Extreme re-SuperHyperDuality, an Extreme v-SuperHyperDuality, and an Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum an Extreme cardinality of the an Extreme SuperHyperEdges of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperDuality;
\item[$(iii)$]
an \textbf{Extreme SuperHyperDuality SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperDuality, an Extreme re-SuperHyperDuality, an Extreme v-SuperHyperDuality, and an 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 conseNeighborive 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{an Extreme SuperHyperDuality SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperDuality, an Extreme re-SuperHyperDuality, an Extreme v-SuperHyperDuality, and an Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the an Extreme SuperHyperPolynomial contains the an Extreme coefficients defined as the an Extreme number of the maximum an Extreme cardinality of the an Extreme SuperHyperEdges of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperDuality; and the an Extreme power is corresponded to its an Extreme coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperDuality} if it's either of an Extreme e-SuperHyperDuality, an Extreme re-SuperHyperDuality, an Extreme v-SuperHyperDuality, and an 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 conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality;
\item[$(vi)$]
a \textbf{an Extreme R-SuperHyperDuality} if it's either of an Extreme e-SuperHyperDuality, an Extreme re-SuperHyperDuality, an Extreme v-SuperHyperDuality, and an Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum an Extreme cardinality of the an Extreme SuperHyperVertices of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperDuality;
\item[$(vii)$]
an \textbf{Extreme R-SuperHyperDuality SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperDuality, an Extreme re-SuperHyperDuality, an Extreme v-SuperHyperDuality, and an 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 conseNeighborive 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{an Extreme SuperHyperDuality SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperDuality, an Extreme re-SuperHyperDuality, an Extreme v-SuperHyperDuality, and an Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the an Extreme SuperHyperPolynomial contains the an Extreme coefficients defined as the an Extreme number of the maximum an Extreme cardinality of the an Extreme SuperHyperVertices of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperDuality; and the an Extreme power is corresponded to its an Extreme coefficient.
\end{itemize}
\end{definition}
\begin{example}\label{136EXM1}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned an Extreme Figures in every an Extreme items.
\begin{itemize}
\item On the Figure \eqref{136NSHG1}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward. $E_1$ and $E_3$ are some empty Extreme
SuperHyperEdges but $E_2$ is a loop an Extreme SuperHyperEdge and $E_4$ is a an Extreme SuperHyperEdge. Thus in the terms of an Extreme SuperHyperNeighbor, there's only one an Extreme SuperHyperEdge, namely, $E_4.$ The an Extreme SuperHyperVertex, $V_3$ is an Extreme isolated means that there's no an Extreme SuperHyperEdge has it as a an Extreme endpoint. Thus the an Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given an Extreme SuperHyperDuality.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG2}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward. $E_1,E_2$ and $E_3$ are some empty an Extreme SuperHyperEdges but $E_4$ is a an Extreme SuperHyperEdge. Thus in the terms of an Extreme SuperHyperNeighbor, there's only one an Extreme SuperHyperEdge, namely, $E_4.$ The an Extreme SuperHyperVertex, $V_3$ is an Extreme isolated means that there's no an Extreme SuperHyperEdge has it as a an Extreme endpoint. Thus the an Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given an Extreme SuperHyperDuality.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG3}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG4}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_4,E_2\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=2z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_1,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG5}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_3\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=4z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_5\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG6}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_{{3i+1}_{i=0}^3},E_{{3i+24}_{i=0}^3}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}6z^8.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_{{3i+1}_{i=0}^7}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=6z^8.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG7}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_{15},E_{16},E_{17}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=z^3.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_3,V_{13},V_8\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
\\&&
4\times5\times5 z^3.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG8}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_3,V_{13},V_8\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
\\&&
4\times5\times5 z^3.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG9}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_{{3i+1}_{i=0}^3},E_{23}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=3z^5.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_{{3i+1}_{i=0}^3},V_{15}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=3z^5.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG10}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_3,V_{13},V_8\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
\\&&
4\times5\times5z^3.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG11}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_1,E_3\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_6,V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
\\&&
3\times3z^2.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG12}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_1\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_1,V^{i\neq 5,7,8}_{i_{i=4}^{10}}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
5z^5.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG13}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_5,E_9\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_1,V_6\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
\\&&
3\times3 z^2.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG14}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_1\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=2z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG15}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_1,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG16}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_1,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
\\&&
(2\times1\times2)+(2\times4\times5)z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG17}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_1,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
\\&&
(1\times1\times2)z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG18}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_1,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
\\&&
(2\times2\times2)z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG19}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_{{3i+1}_{i=0^3}}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=3z^4.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_{{2i+1}_{i=0^5}}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=
2z^6.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG20}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_6\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=10z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=z.
\end{eqnarray*}
\item On the Figure \eqref{95NHG1}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_2\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=2z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=10z.
\end{eqnarray*}
\item On the Figure \eqref{95NHG2}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperDuality, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=\{E_3,E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}=4z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}=\{V_3,V_6\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}
\\&&
=10\times9+10\times6+12\times9+12\times6z^2.
\end{eqnarray*}
\end{itemize}
\end{example}
The previous an Extreme approach apply on the upcoming an Extreme results on an Extreme SuperHyperClasses.
\begin{proposition}
Assume a connected an Extreme SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=
\\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}
\\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}
\\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}
\\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{an 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{an Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\end{eqnarray*}
be a longest path taken from a connected an 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 an Extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The an Extreme SuperHyperSet, in the an Extreme SuperHyperModel \eqref{136NSHG18a}, is the SuperHyperDuality.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality}}=
\\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperDuality SuperHyperPolynomial}}
\\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperDuality}}
\\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperDuality SuperHyperPolynomial}}}
\\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{an 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{an Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\end{eqnarray*}
be a longest path taken from a connected an 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 an Extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained an Extreme SuperHyperSet, in the an Extreme SuperHyperModel \eqref{136NSHG19a},
is the an Extreme SuperHyperDuality.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperStar $ESHS:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperDuality}}=\{E\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperDuality SuperHyperPolynomial}}
\\&&=|i~|~E_i\in E_{ESHG:(V,E)|_{\text{an Extreme Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperDuality}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an 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 an 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 an Extreme SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperStar $ESHS:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG20a}, is the an Extreme SuperHyperDuality.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an 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{an 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{an 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{an Extreme Quasi-SuperHyperDuality SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{an 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 an 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 an Extreme SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-an 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 an Extreme Figure \eqref{136NSHG21a}, the connected an Extreme SuperHyperBipartite $ESHB:(V,E),$ is an Extreme highlighted and an Extreme featured. The obtained an Extreme SuperHyperSet, by the an Extreme Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperBipartite $ESHB:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG21a}, is the an Extreme SuperHyperDuality.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an 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{an 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{an 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{an Extreme Quasi-SuperHyperDuality SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{an 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 an 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 an Extreme SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-an 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 an Extreme SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, the connected an Extreme SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and an Extreme featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperMultipartite $ESHM:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG22a}, is the an Extreme SuperHyperDuality.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperDuality}}=\{E^{*}\in E^{*}_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperDuality SuperHyperPolynomial}}
\\&&=|i~|~E^{*}_i\in E^{*}_{ESHG:(V,E)|_{\text{an Extreme Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperDuality}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an 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{an Extreme Cardinality}}},V^{EXTERNAL}_{|E^{*}_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}+1}
\end{eqnarray*}
is a longest SuperHyperDuality taken from a connected an 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 an Extreme Figure \eqref{136NSHG23a}, the connected an Extreme SuperHyperWheel $NSHW:(V,E),$ is an Extreme highlighted and featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperWheel $ESHW:(V,E),$
in the an Extreme SuperHyperModel \eqref{136NSHG23a}, is the an Extreme SuperHyperDuality.
\end{example}
\section{
an Extreme SuperHyperJoin But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Different an Extreme Types of an Extreme SuperHyperJoin).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a an 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{an 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{an 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{an 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{an 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{an Extreme SuperHyperJoin} if it's either of an Extreme e-SuperHyperJoin, an Extreme re-SuperHyperJoin, an Extreme v-SuperHyperJoin, and an Extreme rv-SuperHyperJoin.
\end{itemize}
\end{definition}
\begin{definition}((an Extreme) SuperHyperJoin).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a an 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 an Extreme e-SuperHyperJoin, an Extreme re-SuperHyperJoin, an Extreme v-SuperHyperJoin, and an 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 conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin;
\item[$(ii)$]
a \textbf{an Extreme SuperHyperJoin} if it's either of an Extreme e-SuperHyperJoin, an Extreme re-SuperHyperJoin, an Extreme v-SuperHyperJoin, and an Extreme rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum an Extreme cardinality of the an Extreme SuperHyperEdges of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperJoin;
\item[$(iii)$]
an \textbf{Extreme SuperHyperJoin SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperJoin, an Extreme re-SuperHyperJoin, an Extreme v-SuperHyperJoin, and an 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 conseNeighborive 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{an Extreme SuperHyperJoin SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperJoin, an Extreme re-SuperHyperJoin, an Extreme v-SuperHyperJoin, and an Extreme rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the an Extreme SuperHyperPolynomial contains the an Extreme coefficients defined as the an Extreme number of the maximum an Extreme cardinality of the an Extreme SuperHyperEdges of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperJoin; and the an Extreme power is corresponded to its an Extreme coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperJoin} if it's either of an Extreme e-SuperHyperJoin, an Extreme re-SuperHyperJoin, an Extreme v-SuperHyperJoin, and an 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 conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperJoin;
\item[$(vi)$]
a \textbf{an Extreme R-SuperHyperJoin} if it's either of an Extreme e-SuperHyperJoin, an Extreme re-SuperHyperJoin, an Extreme v-SuperHyperJoin, and an Extreme rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum an Extreme cardinality of the an Extreme SuperHyperVertices of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperJoin;
\item[$(vii)$]
an \textbf{Extreme R-SuperHyperJoin SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperJoin, an Extreme re-SuperHyperJoin, an Extreme v-SuperHyperJoin, and an 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 conseNeighborive 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{an Extreme SuperHyperJoin SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperJoin, an Extreme re-SuperHyperJoin, an Extreme v-SuperHyperJoin, and an Extreme rv-SuperHyperJoin and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the an Extreme SuperHyperPolynomial contains the an Extreme coefficients defined as the an Extreme number of the maximum an Extreme cardinality of the an Extreme SuperHyperVertices of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperJoin; and the an Extreme power is corresponded to its an Extreme coefficient.
\end{itemize}
\end{definition}
\begin{example}\label{136EXM1}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned an Extreme Figures in every an Extreme items.
\begin{itemize}
\item On the Figure \eqref{136NSHG1}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward. $E_1$ and $E_3$ are some empty an Extreme
SuperHyperEdges but $E_2$ is a loop an Extreme SuperHyperEdge and $E_4$ is a an Extreme SuperHyperEdge. Thus in the terms of an Extreme SuperHyperNeighbor, there's only one an Extreme SuperHyperEdge, namely, $E_4.$ The an Extreme SuperHyperVertex, $V_3$ is an Extreme isolated means that there's no an Extreme SuperHyperEdge has it as a an Extreme endpoint. Thus the an Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given an Extreme SuperHyperJoin.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_1,V_2,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=z^3.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG2}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward. $E_1,E_2$ and $E_3$ are some empty an Extreme SuperHyperEdges but $E_4$ is a an Extreme SuperHyperEdge. Thus in the terms of an Extreme SuperHyperNeighbor, there's only one an Extreme SuperHyperEdge, namely, $E_4.$ The an Extreme SuperHyperVertex, $V_3$ is an Extreme isolated means that there's no an Extreme SuperHyperEdge has it as a an Extreme endpoint. Thus the an Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given an Extreme SuperHyperJoin.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_1,V_2,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=z^3.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG3}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_1,V_2,V_3\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=z^3.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG4}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_4,E_2\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=2z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_1,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG5}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_3\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=4z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_5\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG6}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_{{3i+1}_{i=0}^3},E_{{3i+24}_{i=0}^3}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}6z^8.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_{{3i+1}_{i=0}^7}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=6z^8.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG7}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_{15},E_{16},E_{17}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=z^3.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_3,V_{13},V_8\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
4\times5\times5 z^3.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG8}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_3,V_{13},V_8\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
4\times5\times5 z^3.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG9}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_{{3i+1}_{i=0}^3},E_{23}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^5.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_{{3i+1}_{i=0}^3},V_{15}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=3z^5.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG10}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_3,V_{13},V_8\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
4\times5\times5z^3.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG11}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_1,E_3\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_6,V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
3\times3z^2.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG12}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_1\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_1,V^{i\neq 5,7,8}_{i_{i=4}^{10}}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
5z^5.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG13}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_3,E_9\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_1,V_6\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
3\times3 z^2.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG14}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_1\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=2z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG15}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_1,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG16}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_2,V_7,V_{17}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
(1\times5\times5)+(1\times2+1)z^3.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG17}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_{27},V_2,V_7,V_{17}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
(1\times1\times2+1)z^4.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG18}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_{27},V_2,V_7,V_{17}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
\\&&
(1\times1\times2+1)z^4.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG19}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_{{3i+1}_{i=0^3}}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^4.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_{{2i+1}_{i=0^5}}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=
2z^6.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG20}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_6\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=10z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=z.
\end{eqnarray*}
\item On the Figure \eqref{95NHG1}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_2\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=2z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}=10z.
\end{eqnarray*}
\item On the Figure \eqref{95NHG2}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperJoin, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=\{E_2,E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}=\{V_3,V_6\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}
\\&&
=10\times6+10\times6+12\times6+12\times6z^2.
\end{eqnarray*}
\end{itemize}
\end{example}
The previous an Extreme approach apply on the upcoming an Extreme results on an Extreme SuperHyperClasses.
\begin{proposition}
Assume a connected an Extreme SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=
\\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}
\\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}
\\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}
\\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{an 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{an Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\end{eqnarray*}
be a longest path taken from a connected an 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 an Extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The an Extreme SuperHyperSet, in the an Extreme SuperHyperModel \eqref{136NSHG18a}, is the SuperHyperJoin.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=
\\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}
\\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}
\\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}
\\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{an 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{an Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\end{eqnarray*}
be a longest path taken from a connected an 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 an Extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained an Extreme SuperHyperSet, in the an Extreme SuperHyperModel \eqref{136NSHG19a},
is the an Extreme SuperHyperJoin.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperStar $ESHS:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperJoin}}=\{E\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperJoin SuperHyperPolynomial}}
\\&&=|i~|~E_i\in E_{ESHG:(V,E)|_{\text{an Extreme Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperJoin}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an 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 an 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 an Extreme SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperStar $ESHS:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG20a}, is the an Extreme SuperHyperJoin.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an 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{an 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{an 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{an Extreme SuperHyperJoin SuperHyperPolynomial}}
\\&&
=\text{(OTHERWISE)}0.
\\&&
\mathcal{C}(NSHG)_{\text{an 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{an Extreme Quasi-SuperHyperJoin SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{an 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 an 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 an Extreme SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-an 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 an Extreme Figure \eqref{136NSHG21a}, the connected an Extreme SuperHyperBipartite $ESHB:(V,E),$ is an Extreme highlighted and an Extreme featured. The obtained an Extreme SuperHyperSet, by the an Extreme Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperBipartite $ESHB:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG21a}, is the an Extreme SuperHyperJoin.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an 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{an 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{an 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{an Extreme SuperHyperJoin SuperHyperPolynomial}}
\\&&
=\text{(OTHERWISE)}0.
\\&&
\mathcal{C}(NSHG)_{\text{an 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{an Extreme Quasi-SuperHyperJoin SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{an 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 an 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 an Extreme SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-an 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 an Extreme SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, the connected an Extreme SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and an Extreme featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperMultipartite $ESHM:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG22a}, is the an Extreme SuperHyperJoin.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin}}=
\\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperJoin SuperHyperPolynomial}}
\\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperJoin}}
\\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperJoin SuperHyperPolynomial}}}
\\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{an 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{an Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\end{eqnarray*}
is a longest SuperHyperJoin taken from a connected an 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 an Extreme Figure \eqref{136NSHG23a}, the connected an Extreme SuperHyperWheel $NSHW:(V,E),$ is an Extreme highlighted and featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperWheel $ESHW:(V,E),$
in the an Extreme SuperHyperModel \eqref{136NSHG23a}, is the an Extreme SuperHyperJoin.
\end{example}
\section{
an Extreme SuperHyperPerfect But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Different an Extreme Types of an Extreme SuperHyperPerfect).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a an 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{an 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{an 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{an 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{an 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{an Extreme SuperHyperPerfect} if it's either of an Extreme e-SuperHyperPerfect, an Extreme re-SuperHyperPerfect, an Extreme v-SuperHyperPerfect, and an Extreme rv-SuperHyperPerfect.
\end{itemize}
\end{definition}
\begin{definition}((an Extreme) SuperHyperPerfect).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a an 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 an Extreme e-SuperHyperPerfect, an Extreme re-SuperHyperPerfect, an Extreme v-SuperHyperPerfect, and an 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 conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperPerfect;
\item[$(ii)$]
a \textbf{an Extreme SuperHyperPerfect} if it's either of an Extreme e-SuperHyperPerfect, an Extreme re-SuperHyperPerfect, an Extreme v-SuperHyperPerfect, and an Extreme rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum an Extreme cardinality of the an Extreme SuperHyperEdges of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperPerfect;
\item[$(iii)$]
an \textbf{Extreme SuperHyperPerfect SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperPerfect, an Extreme re-SuperHyperPerfect, an Extreme v-SuperHyperPerfect, and an 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 conseNeighborive 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{an Extreme SuperHyperPerfect SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperPerfect, an Extreme re-SuperHyperPerfect, an Extreme v-SuperHyperPerfect, and an Extreme rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the an Extreme SuperHyperPolynomial contains the an Extreme coefficients defined as the an Extreme number of the maximum an Extreme cardinality of the an Extreme SuperHyperEdges of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperPerfect; and the an Extreme power is corresponded to its an Extreme coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperPerfect} if it's either of an Extreme e-SuperHyperPerfect, an Extreme re-SuperHyperPerfect, an Extreme v-SuperHyperPerfect, and an 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 conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperPerfect;
\item[$(vi)$]
a \textbf{an Extreme R-SuperHyperPerfect} if it's either of an Extreme e-SuperHyperPerfect, an Extreme re-SuperHyperPerfect, an Extreme v-SuperHyperPerfect, and an Extreme rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum an Extreme cardinality of the an Extreme SuperHyperVertices of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperPerfect;
\item[$(vii)$]
an \textbf{Extreme R-SuperHyperPerfect SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperPerfect, an Extreme re-SuperHyperPerfect, an Extreme v-SuperHyperPerfect, and an 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 conseNeighborive 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{an Extreme SuperHyperPerfect SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperPerfect, an Extreme re-SuperHyperPerfect, an Extreme v-SuperHyperPerfect, and an Extreme rv-SuperHyperPerfect and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the an Extreme SuperHyperPolynomial contains the an Extreme coefficients defined as the an Extreme number of the maximum an Extreme cardinality of the an Extreme SuperHyperVertices of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperPerfect; and the an Extreme power is corresponded to its an Extreme coefficient.
\end{itemize}
\end{definition}
\begin{example}\label{136EXM1}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned an Extreme Figures in every an Extreme items.
\begin{itemize}
\item On the Figure \eqref{136NSHG1}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward. $E_1$ and $E_3$ are some empty an Extreme
SuperHyperEdges but $E_2$ is a loop an Extreme SuperHyperEdge and $E_4$ is a an Extreme SuperHyperEdge. Thus in the terms of an Extreme SuperHyperNeighbor, there's only one an Extreme SuperHyperEdge, namely, $E_4.$ The an Extreme SuperHyperVertex, $V_3$ is an Extreme isolated means that there's no an Extreme SuperHyperEdge has it as a an Extreme endpoint. Thus the an Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given an Extreme SuperHyperPerfect.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG2}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward. $E_1,E_2$ and $E_3$ are some empty an Extreme SuperHyperEdges but $E_4$ is a an Extreme SuperHyperEdge. Thus in the terms of an Extreme SuperHyperNeighbor, there's only one an Extreme SuperHyperEdge, namely, $E_4.$ The an Extreme SuperHyperVertex, $V_3$ is an Extreme isolated means that there's no an Extreme SuperHyperEdge has it as a an Extreme endpoint. Thus the an Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given an Extreme SuperHyperPerfect.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG3}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG4}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_4,E_2\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=2z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_1,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG5}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_3\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=4z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_5\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG6}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_{{3i+1}_{i=0}^3},E_{{3i+24}_{i=0}^3}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}6z^8.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_{{3i+1}_{i=0}^7}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=6z^8.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG7}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_{15},E_{16},E_{17}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=z^3.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_3,V_6,V_8\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
\\&&
3\times4\times4 z^3.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG8}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_3,V_6,V_8\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
\\&&
3\times4\times4 z^3.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG9}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperPerfect}}=\{E_{{3i+1}_{i=0}^3},E_{23}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperPerfect SuperHyperPolynomial}}=3z^5.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperPerfect}}=\{V_{{3i+1}_{i=0}^3},V_{15}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=3z^5.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG10}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_3,V_6,V_8\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
\\&&
3\times4\times4z^3.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG11}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_1,E_3\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_6,V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
\\&&
3\times2z^2.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG12}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_1\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_1,V^{i\neq 5,7,8}_{i_{i=4}^{10}}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
5z^5.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG13}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_3,E_9\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_1,V_6\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
\\&&
3\times3 z^2.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG14}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_1\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=2z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG15}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_1,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=
z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG16}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperPerfect}}=\{V_2,V_7,V_{17}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=
\\&&
(1\times5\times5)+(1\times2+1)z^3.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG17}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperPerfect}}=\{V_{27},V_2,V_7,V_{17}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=
\\&&
(1\times1\times2+1)z^4.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG18}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=3z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperPerfect}}=\{V_{27},V_2,V_7,V_{17}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=
\\&&
(1\times1\times2+1)z^4.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG19}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_{{3i+1}_{i=0^3}}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=3z^4.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperPerfect}}=\{V_{{2i+1}_{i=0^5}}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}=
2z^6.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG20}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_6\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=10z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=z.
\end{eqnarray*}
\item On the Figure \eqref{95NHG1}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_2\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=2z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}=10z.
\end{eqnarray*}
\item On the Figure \eqref{95NHG2}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperPerfect, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E_2,E_5\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}=z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperPerfect}}=\{V_3,V_6\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperPerfect SuperHyperPolynomial}}}
\\&&
=10\times6+10\times6+12\times6+12\times6z^2.
\end{eqnarray*}
\end{itemize}
\end{example}
The previous an Extreme approach apply on the upcoming an Extreme results on an Extreme SuperHyperClasses.
\begin{proposition}
Assume a connected an Extreme SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=
\\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}
\\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}
\\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}
\\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{an 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{an Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\end{eqnarray*}
be a longest path taken from a connected an 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 an Extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The an Extreme SuperHyperSet, in the an Extreme SuperHyperModel \eqref{136NSHG18a}, is the SuperHyperPerfect.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=
\\&&=\{E_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}
\\&&=3z^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}
\\&&=\{V^{EXTERNAL}_i\}_{i=1}^{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperPerfect SuperHyperPolynomial}}}
\\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}
z^{\frac{|E_{ESHG:(V,E)}|_{\text{an 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{an Extreme Cardinality}}}{3}},{V^{EXTERNAL}}_{\frac{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}}{3}}.
\end{eqnarray*}
be a longest path taken from a connected an 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 an Extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained an Extreme SuperHyperSet, in the an Extreme SuperHyperModel \eqref{136NSHG19a},
is the an Extreme SuperHyperPerfect.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperStar $ESHS:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}
\\&&=|i~|~E_i\in E_{ESHG:(V,E)|_{\text{an Extreme Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an 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 an 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 an Extreme SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperStar $ESHS:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG20a}, is the an Extreme SuperHyperPerfect.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an 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{an 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{an 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{an Extreme SuperHyperPerfect SuperHyperPolynomial}}
\\&&
=\text{(OTHERWISE)}0.
\\&&
\mathcal{C}(NSHG)_{\text{an 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{an Extreme Quasi-SuperHyperPerfect SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{an 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 an 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 an Extreme SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-an 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 an Extreme Figure \eqref{136NSHG21a}, the connected an Extreme SuperHyperBipartite $ESHB:(V,E),$ is an Extreme highlighted and an Extreme featured. The obtained an Extreme SuperHyperSet, by the an Extreme Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperBipartite $ESHB:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG21a}, is the an Extreme SuperHyperPerfect.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an 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{an 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{an 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{an Extreme SuperHyperPerfect SuperHyperPolynomial}}
\\&&
=\text{(OTHERWISE)}0.
\\&&
\mathcal{C}(NSHG)_{\text{an 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{an Extreme Quasi-SuperHyperPerfect SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{an 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 an 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 an Extreme SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-an 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 an Extreme SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, the connected an Extreme SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and an Extreme featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperMultipartite $ESHM:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG22a}, is the an Extreme SuperHyperPerfect.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect}}=\{E\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperPerfect SuperHyperPolynomial}}
\\&&=|i~|~E_i\in E_{ESHG:(V,E)|_{\text{an Extreme Cardinality}}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperPerfect}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an 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 an 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 an Extreme Figure \eqref{136NSHG23a}, the connected an Extreme SuperHyperWheel $NSHW:(V,E),$ is an Extreme highlighted and featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperWheel $ESHW:(V,E),$
in the an Extreme SuperHyperModel \eqref{136NSHG23a}, is the an Extreme SuperHyperPerfect.
\end{example}
\section{
an Extreme SuperHyperTotal But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Different an Extreme Types of an Extreme SuperHyperTotal).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a an 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{an 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{an 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{an 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{an 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{an Extreme SuperHyperTotal} if it's either of an Extreme e-SuperHyperTotal, an Extreme re-SuperHyperTotal, an Extreme v-SuperHyperTotal, and an Extreme rv-SuperHyperTotal.
\end{itemize}
\end{definition}
\begin{definition}((an Extreme) SuperHyperTotal).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a an 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 an Extreme e-SuperHyperTotal, an Extreme re-SuperHyperTotal, an Extreme v-SuperHyperTotal, and an 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 conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTotal;
\item[$(ii)$]
a \textbf{an Extreme SuperHyperTotal} if it's either of an Extreme e-SuperHyperTotal, an Extreme re-SuperHyperTotal, an Extreme v-SuperHyperTotal, and an Extreme rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum an Extreme cardinality of the an Extreme SuperHyperEdges of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperTotal;
\item[$(iii)$]
an \textbf{Extreme SuperHyperTotal SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperTotal, an Extreme re-SuperHyperTotal, an Extreme v-SuperHyperTotal, and an 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 conseNeighborive 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{an Extreme SuperHyperTotal SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperTotal, an Extreme re-SuperHyperTotal, an Extreme v-SuperHyperTotal, and an Extreme rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the an Extreme SuperHyperPolynomial contains the an Extreme coefficients defined as the an Extreme number of the maximum an Extreme cardinality of the an Extreme SuperHyperEdges of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperTotal; and the an Extreme power is corresponded to its an Extreme coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperTotal} if it's either of an Extreme e-SuperHyperTotal, an Extreme re-SuperHyperTotal, an Extreme v-SuperHyperTotal, and an 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 conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperTotal;
\item[$(vi)$]
a \textbf{an Extreme R-SuperHyperTotal} if it's either of an Extreme e-SuperHyperTotal, an Extreme re-SuperHyperTotal, an Extreme v-SuperHyperTotal, and an Extreme rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum an Extreme cardinality of the an Extreme SuperHyperVertices of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperTotal;
\item[$(vii)$]
an \textbf{Extreme R-SuperHyperTotal SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperTotal, an Extreme re-SuperHyperTotal, an Extreme v-SuperHyperTotal, and an 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 conseNeighborive 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{an Extreme SuperHyperTotal SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperTotal, an Extreme re-SuperHyperTotal, an Extreme v-SuperHyperTotal, and an Extreme rv-SuperHyperTotal and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the an Extreme SuperHyperPolynomial contains the an Extreme coefficients defined as the an Extreme number of the maximum an Extreme cardinality of the an Extreme SuperHyperVertices of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperTotal; and the an Extreme power is corresponded to its an Extreme coefficient.
\end{itemize}
\end{definition}
\begin{example}\label{136EXM1}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned an Extreme Figures in every an Extreme items.
\begin{itemize}
\item On the Figure \eqref{136NSHG1}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an Extremely straightforward. $E_1$ and $E_3$ are some empty an Extreme
SuperHyperEdges but $E_2$ is a loop an Extreme SuperHyperEdge and $E_4$ is a an Extreme SuperHyperEdge. Thus in the terms of an Extreme SuperHyperNeighbor, there's only one an Extreme SuperHyperEdge, namely, $E_4.$ The an Extreme SuperHyperVertex, $V_3$ is an Extreme isolated means that there's no an Extreme SuperHyperEdge has it as a an Extreme endpoint. Thus the an Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given an Extreme SuperHyperTotal.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperTotal}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperTotal}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG2}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an Extremely straightforward. $E_1,E_2$ and $E_3$ are some empty an Extreme SuperHyperEdges but $E_4$ is a an Extreme SuperHyperEdge. Thus in the terms of an Extreme SuperHyperNeighbor, there's only one an Extreme SuperHyperEdge, namely, $E_4.$ The an Extreme SuperHyperVertex, $V_3$ is an Extreme isolated means that there's no an Extreme SuperHyperEdge has it as a an Extreme endpoint. Thus the an Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given an Extreme SuperHyperTotal.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperTotal}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperTotal}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG3}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperTotal}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperTotal}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG4}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-}}=\{E_4,E_2\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}=2z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperTotal}}=\{V_1,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG5}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperTotal}}=\{E_3\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}=4z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperTotal}}=\{V_5\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG6}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal}}=\{E_{{i+1}_{i=0}^{9}}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal SuperHyperPolynomial}}20z^{10}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperTotal}}=\{V_{{i+1}_{i=0}^{9}}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=20z^{10}.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG7}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal}}=\{E_{{i+1}_{i=0}^{9}}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal SuperHyperPolynomial}}10z^{10}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperTotal}}=\{V_{{i+1}_{i=0}^{9}}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=20z^{10}.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG10}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal}}=\{E_{{i+2}_{i=0^{11}}}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal SuperHyperPolynomial}}=11z^{10}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperTotal}}=\{V_{{i+2}_{i=0^{11}}}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperTotal SuperHyperPolynomial}}}=
11z^{10}.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG20}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal}}=\{E_1,E_2\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal SuperHyperPolynomial}}=2z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperTotal}}=\{V_1,V_2\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperTotal SuperHyperPolynomial}}}=9z^2.
\end{eqnarray*}
\item On the Figure \eqref{95NHG2}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperTotal, is up. The an Extreme Algorithm is an 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 an Extreme approach apply on the upcoming an Extreme results on an Extreme SuperHyperClasses.
\begin{proposition}
Assume a connected an Extreme SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal}}=
\\&&=\{E_i\}_{i=1}^{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}-2}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal SuperHyperPolynomial}}
\\&&=z^{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}-2}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperTotal}}
\\&&=\{V^{EXTERNAL}_i\}_{i=1}^{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}-2}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperTotal SuperHyperPolynomial}}}
\\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}
z^{|E_{ESHG:(V,E)}|_{\text{an 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 an 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 an Extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The an Extreme SuperHyperSet, in the an Extreme SuperHyperModel \eqref{136NSHG18a}, is the SuperHyperTotal.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal}}=
\\&&=\{E_i\}_{i=1}^{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}-2}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal SuperHyperPolynomial}}
\\&&=({|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}-1)
\\&&
z^{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}-2}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperTotal}}
\\&&=\{V^{EXTERNAL}_i\}_{i=1}^{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}-2}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperTotal SuperHyperPolynomial}}}
\\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}
z^{|E_{ESHG:(V,E)}|_{\text{an 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{an Extreme Cardinality}}-1}},{V^{EXTERNAL}}_{|E_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}-1}}.
\end{eqnarray*}
be a longest path taken from a connected an 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 an Extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained an Extreme SuperHyperSet, in the an Extreme SuperHyperModel \eqref{136NSHG19a},
is the an Extreme SuperHyperTotal.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperStar $ESHS:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal}}=\{E_i,E_j\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal SuperHyperPolynomial}}
\\&&=|i(i-1)~|~E_i\in E_{ESHG:(V,E)|_{\text{an Extreme Cardinality}}}|z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperTotal}}=\{CENTER,V_j\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperTotal SuperHyperPolynomial}}=
\\&&
(|V_{ESHG:(V,E)|_{\text{an Extreme Cardinality}}}|)~
\text{choose}~
(|V_{ESHG:(V,E)|_{\text{an 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 an 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 an Extreme SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperStar $ESHS:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG20a}, is the an Extreme SuperHyperTotal.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an 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{an Extreme SuperHyperTotal}}
\\&&
\mathcal{C}(NSHG)_{\text{an 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{an 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{an Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{an 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 an 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 an Extreme SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-an 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 an Extreme Figure \eqref{136NSHG21a}, the connected an Extreme SuperHyperBipartite $ESHB:(V,E),$ is an Extreme highlighted and an Extreme featured. The obtained an Extreme SuperHyperSet, by the an Extreme Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperBipartite $ESHB:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG21a}, is the an Extreme SuperHyperTotal.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an 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{an Extreme SuperHyperTotal}}
\\&&
\mathcal{C}(NSHG)_{\text{an 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{an 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{an Extreme Quasi-SuperHyperTotal SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{an 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 an 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 an Extreme SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-an 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 an Extreme SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, the connected an Extreme SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and an Extreme featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperMultipartite $ESHM:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG22a}, is the an Extreme SuperHyperTotal.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal}}=\{E_i,E_j\in E^{*}_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperTotal SuperHyperPolynomial}}
\\&&=|i(i-1)~|~E_i\in E^{*}_{ESHG:(V,E)|_{\text{an Extreme Cardinality}}}|z^2.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperTotal}}=\{CENTER,V_j\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperTotal SuperHyperPolynomial}}=
\\&&
(|V_{ESHG:(V,E)|_{\text{an Extreme Cardinality}}}|)~
\text{choose}~
(|V_{ESHG:(V,E)|_{\text{an 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 an 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 an Extreme Figure \eqref{136NSHG23a}, the connected an Extreme SuperHyperWheel $NSHW:(V,E),$ is an Extreme highlighted and featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperWheel $ESHW:(V,E),$
in the an Extreme SuperHyperModel \eqref{136NSHG23a}, is the an Extreme SuperHyperTotal.
\end{example}
\section{
an Extreme SuperHyperConnected But As The Extensions Excerpt From Dense And Super Forms}
\begin{definition}(Different an Extreme Types of an Extreme SuperHyperConnected).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a an 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{an 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{an 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{an 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{an 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{an Extreme SuperHyperConnected} if it's either of an Extreme e-SuperHyperConnected, an Extreme re-SuperHyperConnected, an Extreme v-SuperHyperConnected, and an Extreme rv-SuperHyperConnected.
\end{itemize}
\end{definition}
\begin{definition}((an Extreme) SuperHyperConnected).\\
Assume a an Extreme SuperHyperGraph (NSHG) $S$
is a pair $S=(V,E).$ Consider a an 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 an Extreme e-SuperHyperConnected, an Extreme re-SuperHyperConnected, an Extreme v-SuperHyperConnected, and an 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 conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperConnected;
\item[$(ii)$]
a \textbf{an Extreme SuperHyperConnected} if it's either of an Extreme e-SuperHyperConnected, an Extreme re-SuperHyperConnected, an Extreme v-SuperHyperConnected, and an Extreme rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum an Extreme cardinality of the an Extreme SuperHyperEdges of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperConnected;
\item[$(iii)$]
an \textbf{Extreme SuperHyperConnected SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperConnected, an Extreme re-SuperHyperConnected, an Extreme v-SuperHyperConnected, and an 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 conseNeighborive 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{an Extreme SuperHyperConnected SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperConnected, an Extreme re-SuperHyperConnected, an Extreme v-SuperHyperConnected, and an Extreme rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the an Extreme SuperHyperPolynomial contains the an Extreme coefficients defined as the an Extreme number of the maximum an Extreme cardinality of the an Extreme SuperHyperEdges of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperConnected; and the an Extreme power is corresponded to its an Extreme coefficient;
\item[$(v)$]
an \textbf{Extreme R-SuperHyperConnected} if it's either of an Extreme e-SuperHyperConnected, an Extreme re-SuperHyperConnected, an Extreme v-SuperHyperConnected, and an 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 conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperConnected;
\item[$(vi)$]
a \textbf{an Extreme R-SuperHyperConnected} if it's either of an Extreme e-SuperHyperConnected, an Extreme re-SuperHyperConnected, an Extreme v-SuperHyperConnected, and an Extreme rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum an Extreme cardinality of the an Extreme SuperHyperVertices of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperConnected;
\item[$(vii)$]
an \textbf{Extreme R-SuperHyperConnected SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperConnected, an Extreme re-SuperHyperConnected, an Extreme v-SuperHyperConnected, and an 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 conseNeighborive 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{an Extreme SuperHyperConnected SuperHyperPolynomial} if it's either of an Extreme e-SuperHyperConnected, an Extreme re-SuperHyperConnected, an Extreme v-SuperHyperConnected, and an Extreme rv-SuperHyperConnected and $\mathcal{C}(NSHG)$ for a an Extreme SuperHyperGraph $NSHG:(V,E)$ is the an Extreme SuperHyperPolynomial contains the an Extreme coefficients defined as the an Extreme number of the maximum an Extreme cardinality of the an Extreme SuperHyperVertices of a an Extreme SuperHyperSet $S$ of high an Extreme cardinality conseNeighborive an Extreme SuperHyperEdges and an Extreme SuperHyperVertices such that they form the an Extreme SuperHyperConnected; and the an Extreme power is corresponded to its an Extreme coefficient.
\end{itemize}
\end{definition}
\begin{example}\label{136EXM1}
Assume a an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned an Extreme Figures in every an Extreme items.
\begin{itemize}
\item On the Figure \eqref{136NSHG1}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an Extremely straightforward. $E_1$ and $E_3$ are some empty an Extreme
SuperHyperEdges but $E_2$ is a loop an Extreme SuperHyperEdge and $E_4$ is a an Extreme SuperHyperEdge. Thus in the terms of an Extreme SuperHyperNeighbor, there's only one an Extreme SuperHyperEdge, namely, $E_4.$ The an Extreme SuperHyperVertex, $V_3$ is an Extreme isolated means that there's no an Extreme SuperHyperEdge has it as a an Extreme endpoint. Thus the an Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given an Extreme SuperHyperConnected.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperConnected}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG2}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an Extremely straightforward. $E_1,E_2$ and $E_3$ are some empty an Extreme SuperHyperEdges but $E_4$ is a an Extreme SuperHyperEdge. Thus in the terms of an Extreme SuperHyperNeighbor, there's only one an Extreme SuperHyperEdge, namely, $E_4.$ The an Extreme SuperHyperVertex, $V_3$ is an Extreme isolated means that there's no an Extreme SuperHyperEdge has it as a an Extreme endpoint. Thus the an Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given an Extreme SuperHyperConnected.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperConnected}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG3}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected}}=\{E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperConnected}}=\{V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=3z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG4}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected}}=\{E_1,E_2,E_4\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected SuperHyperPolynomial}}=z^3.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperConnected}}=\{V_1,V_4\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=15z^2.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG5}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected}}=\{E_3\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected SuperHyperPolynomial}}=4z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperConnected}}=\{V_5\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=z.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG6}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperConnected}}=\{E_{{i+1}_{i=0}^{9}}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}20z^{10}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperConnected}}=\{V_{{i+1}_{i=0}^{9}}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=20z^{10}.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG7}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperConnected}}=\{E_{{i+1}_{i=0}^{9}}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}10z^{10}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperConnected}}=\{V_{{i+1}_{i=11}^{19}},V_{22}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=20z^{10}.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG10}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperConnected}}=\{E_{{i+2}_{i=0^{11}}}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}=11z^{10}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperConnected}}=\{V_{{i+2}_{i=0^{11}}}\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}=
11z^{10}.
\end{eqnarray*}
\item On the Figure \eqref{136NSHG20}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an 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 an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an Extremely straightforward.
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected}}=\{E_2\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected SuperHyperPolynomial}}=z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperConnected}}=\{V_1\}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-SuperHyperConnected SuperHyperPolynomial}}}=10z.
\end{eqnarray*}
\item On the Figure \eqref{95NHG2}, the an Extreme SuperHyperNotion, namely, an Extreme SuperHyperConnected, is up. The an Extreme Algorithm is an 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 an Extreme approach apply on the upcoming an Extreme results on an Extreme SuperHyperClasses.
\begin{proposition}
Assume a connected an Extreme SuperHyperPath $ESHP:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperConnected}}=
\\&&=\{E_i\}_{i=1}^{|{E_{ESHG:(V,E)}|}_{\text{an Extreme Cardinality}}-2}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}
\\&&=z^{{|E_{ESHG:(V,E)}|}_{\text{an Extreme Cardinality}}-2}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperConnected}}
\\&&=\{V^{EXTERNAL}_i\}_{i=1}^{{|E_{ESHG:(V,E)}|}_{\text{an Extreme Cardinality}}-2}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}
\\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}
z^{{|E_{ESHG:(V,E)}|}_{\text{an 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{an Extreme Cardinality}}-1},{V^{EXTERNAL}}_{{|{E}_{ESHG:(V,E)}|}_{\text{an Extreme Cardinality}}-1}.
\end{eqnarray*}
be a longest path taken from a connected an 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 an Extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The an Extreme SuperHyperSet, in the an Extreme SuperHyperModel \eqref{136NSHG18a}, is the SuperHyperConnected.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperCycle $ESHC:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperConnected}}=
\\&&=\{E_i\}_{i=1}^{{|E_{ESHG:(V,E)}|}_{\text{an Extreme Cardinality}}-2}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme Quasi-SuperHyperConnected SuperHyperPolynomial}}
\\&&=({|E_{ESHG:(V,E)}|}_{\text{an Extreme Cardinality}}-1)
\\&&
z^{{|E_{ESHG:(V,E)}|}_{\text{an Extreme Cardinality}}-2}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-Quasi-SuperHyperConnected}}
\\&&=\{V^{EXTERNAL}_i\}_{i=1}^{{|E_{ESHG:(V,E)}|}_{\text{an Extreme Cardinality}}-2}.
\\&&
\mathcal{C}(NSHG)_{{\small\text{an Extreme R-Quasi-SuperHyperConnected SuperHyperPolynomial}}}
\\&&
=\prod |{V^{EXTERNAL}}_{ESHG:(V,E)}|_{\text{an Extreme Cardinality}}
z^{{|E_{ESHG:(V,E)}|}_{\text{an 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{an Extreme Cardinality}}-1},{V^{EXTERNAL}}_{{|E_{ESHG:(V,E)}|}_{\text{an Extreme Cardinality}}-1}.
\end{eqnarray*}
be a longest path taken from a connected an 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 an Extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained an Extreme SuperHyperSet, in the an Extreme SuperHyperModel \eqref{136NSHG19a},
is the an Extreme SuperHyperConnected.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperStar $ESHS:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected}}=\{E_i\in E_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected SuperHyperPolynomial}}
\\&&=|i~|~E_i\in {|{E}_{ESHG:(V,E)}|}_{\text{an Extreme Cardinality}}z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme R-SuperHyperConnected}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an 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 an 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 an Extreme SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperStar $ESHS:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG20a}, is the an Extreme SuperHyperConnected.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperBipartite $ESHB:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an 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{an 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{an 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{an Extreme V-SuperHyperConnected SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{an 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 an 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 an Extreme SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-an 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 an Extreme Figure \eqref{136NSHG21a}, the connected an Extreme SuperHyperBipartite $ESHB:(V,E),$ is an Extreme highlighted and an Extreme featured. The obtained an Extreme SuperHyperSet, by the an Extreme Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperBipartite $ESHB:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG21a}, is the an Extreme SuperHyperConnected.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperMultipartite $ESHM:(V,E).$ Then
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an 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{an 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{an 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{an Extreme V-SuperHyperConnected SuperHyperPolynomial}}
\\&&=\sum_{|V^{EXTERNAL}_{ESHG:(V,E)}|_{\text{an 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 an 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 an Extreme SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-an 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 an Extreme SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward.
\end{proof}
\begin{example}\label{136EXM22a}
In the Figure \eqref{136NSHG22a}, the connected an Extreme SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and an Extreme featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous an Extreme result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperMultipartite $ESHM:(V,E),$ in the an Extreme SuperHyperModel \eqref{136NSHG22a}, is the an Extreme SuperHyperConnected.
\end{example}
\begin{proposition}
Assume a connected an Extreme SuperHyperWheel $ESHW:(V,E).$ Then,
\begin{eqnarray*}
&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected}}=\{E_i\in E^{*}_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme SuperHyperConnected SuperHyperPolynomial}}
\\&&=|i~|~E_i\in {|{E^{*}}_{ESHG:(V,E)}|}_{\text{an Extreme Cardinality}}|z.
\\&&
\mathcal{C}(NSHG)_{\text{an Extreme V-SuperHyperConnected}}=\{CENTER\in V_{ESHG:(V,E)}\}.
\\&&
\mathcal{C}(NSHG)_{\text{an 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 an 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 an Extreme Figure \eqref{136NSHG23a}, the connected an Extreme SuperHyperWheel $NSHW:(V,E),$ is an Extreme highlighted and featured. The obtained an Extreme SuperHyperSet, by the Algorithm in previous result, of the an Extreme SuperHyperVertices of the connected an Extreme SuperHyperWheel $ESHW:(V,E),$
in the an Extreme SuperHyperModel \eqref{136NSHG23a}, is the an 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.
\\
The seminal paper and groundbreaking article is titled ``\textit{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{HG1} 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 “\textit{Journal of Mathematical Techniques and Computational Mathematics(JMTCM)}” with ISO abbreviation ``\textit{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 ``\textit{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{HG2} 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 “\textit{Journal of Mathematical Techniques and Computational Mathematics(JMTCM)}” with ISO abbreviation ``\textit{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.
\\
The seminal paper and groundbreaking article is titled ``\textit{A Research on Cancer's Recognition and Neutrosophic Super Hypergraph by Eulerian Super Hyper Cycles and Hamiltonian Sets as Hyper Covering Versus Super separations}'' in \textbf{Ref.} \cite{HG3} 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 Recognition. It's published in prestigious and fancy journal is entitled “\textit{Journal of Mathematical Techniques and Computational Mathematics(JMTCM)}” with ISO abbreviation ``\textit{J Math Techniques Comput Math}'' in volume 2 and issue 3 with pages 136-148. 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 ``\textit{neutrosophic co-degree and neutrosophic degree alongside chromatic numbers in the setting of some classes related to neutrosophic hypergraphs}'' in \textbf{Ref.} \cite{HG93} by Henry Garrett (2023). 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 “\textit{Journal of Current Trends in Computer Science Research (JCTCSR)}” with ISO abbreviation ``\textit{J Curr Trends Comp Sci Res}'' in volume 2 and issue 1 with pages 16-24. The research article studies deeply with choosing neutrosophic hypergraphs instead of neutrosophic SuperHyperGraph. It's the breakthrough toward independent results based on initial background. In some articles are titled ``\textit{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), ``\textit{0049 | (Failed)1-Zero-Forcing Number in Neutrosophic Graphs}'' in \textbf{Ref.} \cite{HG5} by Henry Garrett (2022), ``\textit{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), ``\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}'' in \textbf{Ref.} \cite{HG7} by Henry Garrett (2022), ``\textit{Neutrosophic Version Of Separates Groups Of Cells In Cancer’s Recognition On Neutrosophic SuperHyperGraphs}'' in \textbf{Ref.} \cite{HG8} by Henry Garrett (2022), ``\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}'' in \textbf{Ref.} \cite{HG9} by Henry Garrett (2022), ``\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}'' in \textbf{Ref.} \cite{HG10} by Henry Garrett (2022), ``\textit{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), ``\textit{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), ``\textit{(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs}'' in \textbf{Ref.} \cite{HG13} by Henry Garrett (2022), ``\textit{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), ``\textit{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), ``\textit{(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well- SuperHyperModelled (Neutrosophic) SuperHyperGraphs
}'' in \textbf{Ref.} \cite{HG16} by Henry Garrett (2022), ``\textit{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), ``\textit{Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs}'' in \textbf{Ref.} \cite{HG17} by Henry Garrett (2022), ``\textit{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),``\textit{(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances}'' in \textbf{Ref.} \cite{HG19} by Henry Garrett (2022), ``\textit{(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), ``\textit{SuperHyperGirth on SuperHyperGraph and Neutrosophic SuperHyperGraph With SuperHyperModeling of Cancer’s Recognitions}'' in \textbf{Ref.} \cite{HG21} by Henry Garrett (2022), ``\textit{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), ``\textit{SuperHyperDominating and SuperHyperResolving on Neutrosophic SuperHyperGraphs And Their Directions in Game Theory and Neutrosophic SuperHyperClasses}'' in \textbf{Ref.} \cite{HG23} by Henry Garrett (2022), ``\textit{SuperHyperMatching By (R-)Definitions And Polynomials To Monitor Cancer’s Recognition In Neutrosophic SuperHyperGraphs}'' in \textbf{Ref.} \cite{HG24} by Henry Garrett (2023), ``\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}'' in \textbf{Ref.} \cite{HG25} by Henry Garrett (2023), ``\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}'' in \textbf{Ref.} \cite{HG26} by Henry Garrett (2023), ``\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}'' in \textbf{Ref.} \cite{HG27} by Henry Garrett (2023), ``\textit{Perfect Directions Toward Idealism in Cancer's Neutrosophic Recognition Forwarding Neutrosophic SuperHyperClique on Neutrosophic SuperHyperGraphs}'' in \textbf{Ref.} \cite{HG28} by Henry Garrett (2023), ``\textit{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), ``\textit{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), ``\textit{Using the Tool As (Neutrosophic) Failed SuperHyperStable To SuperHyperModel Cancer's Recognition Titled (Neutrosophic) SuperHyperGraphs}'' in \textbf{Ref.} \cite{HG31} by Henry Garrett (2023), ``\textit{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), ``\textit{(Neutrosophic) SuperHyperStable on Cancer’s Recognition by Well-SuperHyperModelled (Neutrosophic) SuperHyperGraphs}'' in \textbf{Ref.} \cite{HG33} by Henry Garrett (2023), ``\textit{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), ``\textit{(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs}'' in \textbf{Ref.} \cite{HG35} by Henry Garrett (2022), ``\textit{Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs}'' in \textbf{Ref.} \cite{HG36} by Henry Garrett (2022), ``\textit{Basic Neutrosophic Notions Concerning SuperHyperDominating and Neutrosophic SuperHyperResolving in SuperHyperGraph}'' in \textbf{Ref.} \cite{HG37} by Henry Garrett (2022), ``\textit{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{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,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,HG171,HG172,HG173,HG174,HG175,HG176,HG177,HG178,HG179,HG180,HG181,HG182,HG183,HG184,HG185,HG186,HG187,HG188,HG189,HG190,HG191,HG192,HG193,HG194,HG195,HG196,HG197,HG198,HG199,HG200,HG201,HG202,HG203,HG204,HG205,HG206,HG207,HG208,HG209,HG210,HG211,HG212,HG213,HG214,HG215,HG93,HG217,HG218,HG219,HG220,HG221,HG222,HG223,HG224,HG225,HG226,HG228,HG230,HG231,HG232,HG233,HG234,HG235,HG236,HG237,HG238,HG239,HG240,HG241,HG242,HG243,HG244,HG245,HG246,HG247,HG248,HG249,HG250,HG251,HG252,HG253,HG254,HG255,HG256,HG257,HG258,HG259}, there are some endeavors to formalize the basic SuperHyperNotions about neutrosophic SuperHyperGraph and SuperHyperGraph alongside scientific research books at \cite{HG60b,HG61b,HG62b,HG63b,HG64b,HG65b,HG66b,HG67b,HG68b,HG69b,HG70b,HG71b,HG72b,HG73b,HG74b,HG75b,HG76b,HG77b,HG78b,HG79b,HG80b,HG81b,HG82b,HG83b,HG84b,HG85b,HG86b,HG87b,HG88b,HG89b,HG90b,HG91b,HG92b,HG93b,HG94b,HG95b,HG96b,HG97b,HG98b,HG99b,HG100b,HG101b,HG102b,HG103b,HG104b,HG105b,HG106b,HG107b,HG108b,HG109b,HG110b,HG111b,HG112b,HG113b,HG114b,HG115b,HG116b,HG117b,HG118b,HG119b,HG120b,HG121b,HG122b,HG123b,HG124b,HG125b,HG126b,HG127b,HG128b,HG129b,HG130b,HG131b,HG132b,HG133b,HG134b,HG135b,HG136b,HG137b,HG138b,HG139b,HG140b,HG141b,HG142b,HG143b,HG144b,HG145b,HG146b,HG147b,HG148b,HG149b,HG150b,HG151b,HG152b,HG153b,HG154b,HG155b,HG156b,HG157b,HG158b,HG159b,HG160b,HG161b,HG162b,HG163b,HG164b,HG165b,HG166b,HG167b,HG168b}. Two popular scientific research books in Scribd in the terms of high readers, 4190 and 5189 respectively, on neutrosophic science is on \cite{HG32b,HG44b}.
\\
Some scientific studies and scientific researches about neutrosophic graphs, are proposed as book in \textbf{Ref.} \cite{HG71b} by Henry Garrett (2023) which is indexed by Google Scholar and has more than 4331 readers in Scribd. It's titled ``\textit{Beyond Neutrosophic Graphs}'' and published by Dr. Henry Garrett. 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{HG70b} by Henry Garrett (2023) which is indexed by Google Scholar and has more than 5327 readers in Scribd. It's titled ``\textit{Neutrosophic Duality}'' and published by Dr. Henry Garrett. 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 notions in SuperHyperGraphs, Neutrosophic notions 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,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,HG171,HG172,HG173,HG174,HG175,HG176,HG177,HG178,HG179,HG180,HG181,HG182,HG183,HG184,HG185,HG186,HG187,HG188,HG189,HG190,HG191,HG192,HG193,HG194,HG195,HG196,HG197,HG198,HG199,HG200,HG201,HG202,HG203,HG204,HG205,HG206,HG207,HG208,HG209,HG210,HG211,HG212,HG213,HG214,HG215,HG93,HG217,HG218,HG219,HG220,HG221,HG222,HG223,HG224,HG225,HG226,HG228,HG230,HG231,HG232,HG233,HG234,HG235,HG236,HG237,HG238,HG239,HG240,HG241,HG242,HG243,HG244,HG245,HG246,HG247,HG248,HG249,HG250,HG251,HG252,HG253,HG254,HG255,HG256,HG257,HG258,HG259} alongside scientific research books at \cite{HG60b,HG61b,HG62b,HG63b,HG64b,HG65b,HG66b,HG67b,HG68b,HG69b,HG70b,HG71b,HG72b,HG73b,HG74b,HG75b,HG76b,HG77b,HG78b,HG79b,HG80b,HG81b,HG82b,HG83b,HG84b,HG85b,HG86b,HG87b,HG88b,HG89b,HG90b,HG91b,HG92b,HG93b,HG94b,HG95b,HG96b,HG97b,HG98b,HG99b,HG100b,HG101b,HG102b,HG103b,HG104b,HG105b,HG106b,HG107b,HG108b,HG109b,HG110b,HG111b,HG112b,HG113b,HG114b,HG115b,HG116b,HG117b,HG118b,HG119b,HG120b,HG121b,HG122b,HG123b,HG124b,HG125b,HG126b,HG127b,HG128b,HG129b,HG130b,HG131b,HG132b,HG133b,HG134b,HG135b,HG136b,HG137b,HG138b,HG139b,HG140b,HG141b,HG142b,HG143b,HG144b,HG145b,HG146b,HG147b,HG148b,HG149b,HG150b,HG151b,HG152b,HG153b,HG154b,HG155b,HG156b,HG157b,HG158b,HG159b,HG160b,HG161b,HG162b,HG163b,HG164b,HG165b,HG166b,HG167b,HG168b}. Two popular scientific research books in Scribd in the terms of high readers, 4331 and 5327 respectively, on neutrosophic science is on \cite{HG32b,HG44b}.
\begin{thebibliography}{595}
\bibitem{HG1} Henry Garrett, “\textit{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{HG2} Henry Garrett, “\textit{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{HG3} Henry Garrett, “\textit{A Research on Cancer's Recognition and Neutrosophic Super Hypergraph by Eulerian Super Hyper Cycles and Hamiltonian Sets as Hyper Covering Versus Super separations}”, J Math Techniques Comput Math 2(3) (2023) 136-148. (https://www.opastpublishers.com/open-access-articles/a-research-on-cancers-recognition-and-neutrosophic-super-hypergraph-by-eulerian-super-hyper-cycles-and-hamiltonian-sets-.pdf)
\bibitem{HG93} 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 2(1) (2023) 16-24. (https://www.opastpublishers.com/open-access-articles/neutrosophic-codegree-and-neutrosophic-degree-alongside-chromatic-numbers-in-the-setting-of-some-classes-related-to-neut.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{``\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{``\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{``\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{HG259}
Henry Garrett, “New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Eulerian-Type-Cycle-Cut As Hyper Nebbish On Super Nebulous”, Zenodo 2023, (doi: 10.5281/zenodo.7871026).
\bibitem{HG258}
Henry Garrett, “New Ideas On Super Nebulous By Hyper Nebbish Of Eulerian-Type-Cycle-Cut In Recognition of Cancer With (Neutrosophic) SuperHyperGraph”, Zenodo 2023, (doi: 10.5281/zenodo.7874647).
\bibitem{HG257}
Henry Garrett, “New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Eulerian-Cycle-Neighbor As Hyper Nebbish On Super Nebulous”, Zenodo 2023, (doi: 10.5281/zenodo.7857856).
\bibitem{HG256}
Henry Garrett, “New Ideas On Super Nebulous By Hyper Nebbish Of Eulerian-Cycle-Neighbor In Recognition of Cancer With (Neutrosophic) SuperHyperGraph”, Zenodo 2023, (doi: 10.5281/zenodo.7857841).
\bibitem{HG255}
Henry Garrett, “New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Eulerian-Cycle-Decomposition As Hyper Decompress On Super Decompensation”, Zenodo 2023, (doi: 10.5281/zenodo.7855661).
\bibitem{HG254}
Henry Garrett, “New Ideas On Super Decompensation By Hyper Decompress Of Eulerian-Cycle-Decomposition In Recognition of Cancer With (Neutrosophic) SuperHyperGraph”, Zenodo 2023, (doi: 10.5281/zenodo.7855637).
\bibitem{HG253}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Eulerian-Cycle-Cut As Hyper Eulogy On Super EULA}”, Zenodo 2023, (doi: 10.5281/zenodo.7853867).
\bibitem{HG252}
Henry Garrett, “\textit{New Ideas On Super EULA By Hyper Eulogy Of Eulerian-Cycle-Cut In Recognition of Cancer With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7853922).
\bibitem{HG251}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Eulerian-Type-Path-Neighbor As Hyper Nebbish On Super Nebulous}”, Zenodo 2023, (doi: 10.5281/zenodo.7851519).
\bibitem{HG250}
Henry Garrett, “\textit{New Ideas On Super Nebulous By Hyper Nebbish Of Eulerian-Type-Path-Neighbor In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7851550).
\bibitem{HG249}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Eulerian-Type-Path-Decomposition As Hyper Decompress On Super Decompensation}”, Zenodo 2023, (doi: 10.5281/zenodo.7839333).
\bibitem{HG248}
Henry Garrett, “\textit{New Ideas On Super Decompensation By Hyper Decompress Of Eulerian-Type-Path-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7840206).
\bibitem{HG247}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Eulerian-Type-Path-Cut As Hyper Eulogy On Super EULA}”, Zenodo 2023, (doi: 10.5281/zenodo.7834229).
\bibitem{HG246}
Henry Garrett, “\textit{New Ideas On Super EULA By Hyper Eulogy Of Eulerian-Type-Path-Cut In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7834261).
\bibitem{HG245}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Eulerian-Path-Neighbor As Hyper Nebbish On Super Nebulous}”, Zenodo 2023, (doi: 10.5281/zenodo.7824560).
\bibitem{HG244}
Henry Garrett, “\textit{New Ideas On Super Nebulous By Hyper Nebbish Of Eulerian-Path-Neighbor In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7824623).
\bibitem{HG243}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Eulerian-Path-Decomposition As Hyper Decompress On Super Decompensation}”, Zenodo 2023, (doi: 10.5281/zenodo.7819531).
\bibitem{HG242}
Henry Garrett, “\textit{New Ideas On Super Decompensation By Hyper Decompress Of Eulerian-Path-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7819579).
\bibitem{HG241}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph As Hyper Tool On Super Toot}”, Zenodo 2023, (doi: 10.5281/zenodo.7812236).
\bibitem{HG240}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By initial Eulerian-Path-Cut As Hyper initial Eulogy On Super initial EULA}”, Zenodo 2023, (doi: 10.5281/zenodo.7809365).
\bibitem{HG239}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Eulerian-Path-Cut As Hyper Eulogy-Path-Cut On Super EULA-Path-Cut}”, Zenodo 2023, (doi: 10.5281/zenodo.7809358).
\bibitem{HG238}
Henry Garrett, “\textit{New Ideas On Super EULA By Hyper Eulogy Of Eulerian-Path-Cut In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7809219).
\bibitem{HG237}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Eulerian-Path-Cut As Hyper Eulogy On Super EULA}”, Zenodo 2023, (doi: 10.5281/zenodo.7809328).
\bibitem{HG236}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Type-Cycle-Neighbor As Hyper Nebbish On Super Nebulous}”, Zenodo 2023, (doi: 10.5281/zenodo.7806767).
\bibitem{HG235}
Henry Garrett, “\textit{New Ideas On Super Nebulous By Hyper Nebbish Of Hamiltonian-Type-Cycle-Neighbor In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7806838).
\bibitem{HG234}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Type-Cycle-Decomposition As Hyper Decompress On Super Decompensation}”, Zenodo 2023, (doi: 10.5281/zenodo.7804238).
\bibitem{HG233}
Henry Garrett, “\textit{New Ideas On Super Decompensation By Hyper Decompress Of Hamiltonian-Type-Cycle-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7804228).
\bibitem{HG232}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Type-Cycle-Cut As Hyper Hamper On Super Hammy}”, Zenodo 2023, (doi: 10.5281/zenodo.7799902).
\bibitem{HG231}
Henry Garrett, “\textit{New Ideas On Super Hammy By Hyper Hamper Of Hamiltonian-Type-Cycle-Cut In Recognition of Cancer With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7804218).
\bibitem{HG230}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Cycle-Neighbor As Hyper Nebbish On Super Nebulous}”, Zenodo 2023, (doi: 10.5281/zenodo.7796334).
\bibitem{HG228}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Cycle-Decomposition As Hyper Decompress On Super Decompensation}”, Zenodo 2023, (doi: 10.5281/zenodo.7793372).
\bibitem{HG226}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Cycle-Cut As Hyper Hamper On Super Hammy}”, Zenodo 2023, (doi: 10.5281/zenodo.7791952).
\bibitem{HG225}
Henry Garrett, “\textit{New Ideas On Super Hammy By Hyper Hamper Of Hamiltonian-Cycle-Cut In Recognition of Cancer With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7791982).
\bibitem{HG224}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Neighbor As Hyper Nebbish On Super Nebulous}”, Zenodo 2023, (doi: 10.5281/zenodo.7790026).
\bibitem{HG223}
Henry Garrett, “\textit{New Ideas On Super Nebulous By Hyper Nebbish Of Hamiltonian-Neighbor In Recognition of Cancer With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7790052).
\bibitem{HG222}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Decomposition As Hyper Decompress On Super Decompensation}”, Zenodo 2023, (doi: 10.5281/zenodo.7787066).
\bibitem{HG221}
Henry Garrett, “\textit{New Ideas On Super Decompensation By Hyper Decompress Of Hamiltonian-Decomposition In Recognition of Cancer With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7787094).
\bibitem{HG220}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Hamiltonian-Cut As Hyper Hamper On Super Hammy}”, Zenodo 2023, (doi: 10.5281/zenodo.7781476).
\bibitem{HG219}
Henry Garrett, “\textit{New Ideas On Super Hammy By Hyper Hamper Of Hamiltonian-Cut In Recognition of Cancer With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7783082).
\bibitem{HG218}
Henry Garrett, “\textit{New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Trace-Neighbor As Hyper Nebbish On Super Nebulous}”, Zenodo 2023, (doi: 10.5281/zenodo.7777857).
\bibitem{HG217}
Henry Garrett, “\textit{New Ideas On Super Nebulous By Hyper Nebbish Of Trace-Neighbor In Recognition of Cancer With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7779286).
\bibitem{HG215}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Trace-Decomposition As Hyper Decompress On Super Decompensation}”, Zenodo 2023, (doi: 10.5281/zenodo.7771831).
\bibitem{HG214}
Henry Garrett, “\textit{New Ideas On Super Decompensation By Hyper Decompress Of Trace-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7772468).
\bibitem{HG213}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Trace-Cut As Hyper Nebbish On Super Nebulous}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.20913.25446).
\bibitem{HG212}
Henry Garrett, “\textit{New Ideas On Super Tract By Hyper Track Of Trace-Cut In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, Zenodo 2023, (doi: 10.5281/zenodo.7764916).
\bibitem{HG211}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Edge-Neighbor As Hyper Nebbish On Super Nebulous}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.11770.98247).
\bibitem{HG210}
Henry Garrett, “\textit{New Ideas On Super Nebulous By Hyper Nebbish Of Edge-Neighbor In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.12400.12808).
\bibitem{HG209}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Edge-Decomposition As Hyper Decompress On Super Decompensation}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.22545.10089).
\bibitem{HG208}
Henry Garrett, “\textit{New Ideas On Super Decompensation By Hyper Decompress Of Edge-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.29544.34564).
\bibitem{HG207}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Edge-Cut As Hyper Edify On Super Eddy}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.11377.76644).
\bibitem{HG206}
Henry Garrett, “\textit{New Ideas On Super Eddy By Hyper Edify Of Edge-Cut In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.23750.96329).
\bibitem{HG205}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Vertex-Neighbor As Hyper Nebbish On Super Nebulous}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.31366.24641).
\bibitem{HG204}
Henry Garrett, “\textit{New Ideas On Super Nebulous By Hyper Nebbish Of Vertex-Neighbor In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.34721.68960).
\bibitem{HG203}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Vertex-Decomposition As Hyper Decompress On Super Decompensation}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.30212.81289).
\bibitem{HG202}
Henry Garrett, “\textit{New Ideas On Super Decompensation By Hyper Decompress Of Vertex-Decomposition In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.18468.76169).
\bibitem{HG201}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Vertex-Cut As Hyper Vertu On Super Vertigo}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.24288.35842).
\bibitem{HG200}
Henry Garrett, “\textit{New Ideas On Super Vertigo By Hyper Vertu Of Vertex-Cut In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.32467.25124).
\bibitem{HG199}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Stable-Neighbor As Hyper Nebbish On Super Nebulous}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.31025.45925).
\bibitem{HG198}
Henry Garrett, “\textit{New Ideas On Super Nebulizer By Hyper Nub Of Stable-Neighbor In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.17184.25602).
\bibitem{HG197}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Stable-Decompositions As Hyper Stain On Super Stagy}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.23423.28327).
\bibitem{HG196}
Henry Garrett, “\textit{New Ideas On Super Stale By Hyper Stalk Of Stable-Decompositions In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.28456.44805).
\bibitem{HG195}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By
Stable-Cut As Hyper Stain On Super Stagy}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.23525.68320).
\bibitem{HG194}
Henry Garrett, “\textit{New Ideas On Super Stale By Hyper Stalk Of Stable-Cut In Cancer's Recognition
With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.20170.24000).
\bibitem{HG193}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By
Clique-Neighbors As Hyper Nebbish On Super Nebulous}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.36475.59683).
\bibitem{HG192}
Henry Garrett, “\textit{New Ideas On Super Nebulizer By Hyper Nub Of Clique-Neighbors In Cancer's
Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.29764.71046).
\bibitem{HG191}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By
Clique-Decompositions As Hyper Decompile On Super Decommission}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.18780.87683).
\bibitem{HG190}
Henry Garrett, “\textit{New Ideas On Super Decompensation By Hyper Decompress Of Clique-
Decompositions In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.27169.48487).
\bibitem{HG189}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By
Clique-Cut As Hyper Click On Super Cliche}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.26134.01603).
\bibitem{HG188}
Henry Garrett, “\textit{New Ideas On Super Cliff By Hyper Cling Of Clique-Cut In Cancer's Recognition
With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.27392.30721).
\bibitem{HG187}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By
Space As Hyper Spin On Super Spacy}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.33028.40321).
\bibitem{HG186}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By List-
Coloring As Hyper List On Super Lisle}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.21389.20966).
\bibitem{HG185}
Henry Garrett, “\textit{New Ideas On Super Lith By Hyper Lite Of List-Coloring In Cancer's Recognition
With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.16356.04489).
\bibitem{HG184}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Space As Hyper Sparse On Super Spark}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.21756.21129).
\bibitem{HG183}
Henry Garrett, “\textit{New Ideas On Super Solidarity By Hyper Soul Of Space In Cancer's Recognition With (Extreme) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.30983.68009).
\bibitem{HG182}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Edge-Connectivity As Hyper Disclosure On Super Closure}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.28552.29445).
\bibitem{HG181}
Henry Garrett, “\textit{New Ideas On Super Uniform By Hyper Deformation Of Edge-Connectivity In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.10936.21761).
\bibitem{HG180}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Vertex-Connectivity As Hyper Leak On Super Structure}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.35105.89447).
\bibitem{HG179}
Henry Garrett, “\textit{New Ideas On Super System By Hyper Explosions Of Vertex-Connectivity In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.30072.72960).
\bibitem{HG178}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Tree-Decomposition As Hyper Forward On Super Returns}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.31147.52003).
\bibitem{HG177}
Henry Garrett, “\textit{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).
\bibitem{HG176}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By Chord As Hyper Excellence On Super Excess}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.13059.58401).
\bibitem{HG175}
Henry Garrett, “\textit{New Ideas On Super Gap By Hyper Navigations Of Chord In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.11172.14720).
\bibitem{HG174}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By SuperHyper(i,j)-Domination As Hyper Controller On Super Waves}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.22011.80165).
\bibitem{HG173}
Henry Garrett, “\textit{New Ideas On Super Coincidence By Hyper Routes Of SuperHyper(i,j)-Domination In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.30819.84003).
\bibitem{HG172}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By SuperHyperEdge-Domination As Hyper Reversion On Super Indirection}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.10493.84962).
\bibitem{HG171}
Henry Garrett, “\textit{New Ideas On Super Obstacles By Hyper Model Of SuperHyperEdge-Domination In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.13849.29280).
\bibitem{HG170}
Henry Garrett, “\textit{New Ideas In Cancer's Recognition And Neutrosophic SuperHyperGraph By SuperHyperK-Domination As Hyper k-Actions On Super Patterns}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.19944.14086).
\bibitem{HG169}
Henry Garrett, “\textit{New Ideas On Super Harmony By Hyper k-Function Of SuperHyperK-Domination In Cancer's Recognition With (Neutrosophic) SuperHyperGraph}”, ResearchGate 2023, (doi: 10.13140/RG.2.2.23299.58404).
\bibitem{HG168}
Henry Garrett, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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, “\textit{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,“\textit{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,“\textit{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{HG132} Henry Garrett,“\textit{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,“\textit{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,“\textit{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,“\textit{(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,“\textit{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,“\textit{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,“\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{HG125} 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{HG124} 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{HG123} 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{HG122} 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{HG121} 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{HG120} 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{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,“\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{HG115} Henry Garrett, “\textit{(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, “\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{HG111} 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{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, “\textit{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, “\textit{(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, “\textit{(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances}”, Preprints 2022, 2022120549 (doi: 10.20944/preprints202212.0549.v1).
\bibitem{HG98} Henry Garrett, “\textit{(Neutrosophic) SuperHyperModeling of Cancer’s Recognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances}”, ResearchGate 2022, (doi: 10.13140/RG.2.2.19380.94084).
\bibitem{HG972} 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{HG97} 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}”, ResearchGate 2022, (doi: 10.13140/RG.2.2.14426.41923).
\bibitem{HG962} 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{HG96} Henry Garrett, “\textit{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,“\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{HG95} Henry Garrett, “\textit{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, “\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{HG94} Henry Garrett, “\textit{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{``\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{HG168b}
Henry Garrett, “Eulerian-Type-Cycle-Cut In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7874677).
\bibitem{HG167b}
Henry Garrett, “Eulerian-Cycle-Neighbor In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7857906).
\bibitem{HG166b}
Henry Garrett, “Eulerian-Cycle-Decomposition In SuperHyperGraphs”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7856329).
\bibitem{HG165b}
Henry Garrett, “\textit{Eulerian-Cycle-Cut In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7854561).
\bibitem{HG164b}
Henry Garrett, “\textit{Eulerian-Type-Path-Neighbor In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7851893).
\bibitem{HG163b}
Henry Garrett, “\textit{Eulerian-Type-Path-Decomposition In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7848019).
\bibitem{HG162b}
Henry Garrett, “\textit{Eulerian-Type-Path-Cut In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7835063).
\bibitem{HG161b}
Henry Garrett, “\textit{Eulerian-Path-Neighbor In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7826705).
\bibitem{HG160b}
Henry Garrett, “\textit{Eulerian-Path-Decomposition In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7820680).
\bibitem{HG159b}
Henry Garrett, “\textit{Eulerian-Path-Cut In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7812750).
\bibitem{HG158b}
Henry Garrett, “\textit{Hamiltonian-Type-Cycle-Neighbor In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7812142).
\bibitem{HG157b}
Henry Garrett, “\textit{Hamiltonian-Type-Cycle-Decomposition In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7810394).
\bibitem{HG156b}
Henry Garrett, “\textit{Hamiltonian-Type-Cycle-Cut In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7807782).
\bibitem{HG155b}
Henry Garrett, “\textit{Hamiltonian-Cycle-Neighbor In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7804449).
\bibitem{HG154b}
Henry Garrett, “\textit{Hamiltonian-Cycle-Decomposition In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7793875).
\bibitem{HG153b}
Henry Garrett, “\textit{Hamiltonian-Cycle-Cut In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7792307).
\bibitem{HG152b} Henry Garrett, “\textit{Hamiltonian-Neighbor In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7790728).
\bibitem{HG151b} Henry Garrett, “\textit{Hamiltonian-Decomposition In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7787712).
\bibitem{HG150b} Henry Garrett, “\textit{Hamiltonian-Cut In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7783791).
\bibitem{HG149b} Henry Garrett, “\textit{Trace-Neighbor In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7780123).
\bibitem{HG148b} Henry Garrett, “\textit{Trace-Decomposition In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7773119).
\bibitem{HG147b} Henry Garrett, “\textit{SuperHyperDuality}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7637762).
\bibitem{HG146b} Henry Garrett, “\textit{Trace-Cut In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7766174).
\bibitem{HG145b}Henry Garrett, “\textit{Edge-Neighbor In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7762232).
\bibitem{HG144b} Henry Garrett, “\textit{Edge-Decomposition In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7758601).
\bibitem{HG143b} Henry Garrett, “\textit{Edge-Cut In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7754661).
\bibitem{HG142b} Henry Garrett, “\textit{Vertex-Neighbor In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7750995) .
\bibitem{HG141b} Henry Garrett, “\textit{Vertex-Decomposition In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7749875).
\bibitem{HG140b} Henry Garrett, “\textit{Vertex-Cut In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7747236).
\bibitem{HG139b} Henry Garrett, “\textit{Stable-Neighbor In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7742587).
\bibitem{HG138b} Henry Garrett, “\textit{Stable-Decompositions In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7738635).
\bibitem{HG137b} Henry Garrett, “\textit{Stable-Cut In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7734719).
\bibitem{HG136b} Henry Garrett, “\textit{Clique-Neighbors In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7730484).
\bibitem{HG135b} Henry Garrett, “\textit{Clique-Decompositions In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7730469).
\bibitem{HG134b} Henry Garrett, “\textit{Clique-Cut In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7722865).
\bibitem{HG133b} Henry Garrett, “\textit{Space In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7713563).
\bibitem{HG132b} Henry Garrett, “\textit{Space In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7709116).
\bibitem{HG131b} Henry Garrett, “\textit{Edge-Connectivity In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7706415).
\bibitem{HG130b} Henry Garrett, “\textit{Vertex-Connectivity In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7706063).
\bibitem{HG129b} Henry Garrett, “\textit{Tree-Decomposition In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7701906).
\bibitem{HG128b} Henry Garrett, “\textit{Chord In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7700205).
\bibitem{HG127b} Henry Garrett, “\textit{(i,j)-Domination In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7694876).
\bibitem{HG126b} Henry Garrett, “\textit{Edge-Domination In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7679410).
\bibitem{HG125b} Henry Garrett, “\textit{K-Domination In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7675982).
\bibitem{HG124b} Henry Garrett, “\textit{K-Number In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7672388).
\bibitem{HG123b} Henry Garrett, “\textit{Order In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7668648).
\bibitem{HG122b} Henry Garrett, “\textit{Coloring In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7662810).
\bibitem{HG121b} Henry Garrett, “\textit{Dimension In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7659162).
\bibitem{HG120b} Henry Garrett, “\textit{Cancer In SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7653233).
\bibitem{HG119b} Henry Garrett, “\textit{SuperHyperWheel}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7653204).
\bibitem{HG118b} Henry Garrett, “\textit{SuperHyperMultipartite}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7653142).
\bibitem{HG117b} Henry Garrett, “\textit{SuperHyperBipartite}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7653117).
\bibitem{HG116b} Henry Garrett, “\textit{SuperHyperStar}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7653089).
\bibitem{HG115b} Henry Garrett, “\textit{SuperHyperCycle}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7651687).
\bibitem{HG114b} Henry Garrett, “\textit{SuperHyperPath}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7651619).
\bibitem{HG113b} Henry Garrett, “\textit{SuperHyperDomination}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7651439).
\bibitem{HG112b} Henry Garrett, “\textit{SuperHyperDominating}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7650729).
\bibitem{HG111b} Henry Garrett, “\textit{SuperHyperConnected}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7647868).
\bibitem{HG110b} Henry Garrett, “\textit{SuperHyperTotal}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7647017).
\bibitem{HG109b} Henry Garrett, “\textit{SuperHyperPerfect}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7644894).
\bibitem{HG108b} Henry Garrett, “\textit{SuperHyperJoin}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7641880).
\bibitem{HG107b} Henry Garrett, “\textit{Path SuperHyperColoring}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7632923).
\bibitem{HG106b} Henry Garrett, “\textit{SuperHyperDensity}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7623459).
\bibitem{HG105b} Henry Garrett, “\textit{Neutrosophic SuperHyperConnectivities}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7606416).
\bibitem{HG104b} Henry Garrett, “\textit{Extreme SuperHyperConnectivities}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7606416).
\bibitem{HG103b} Henry Garrett, “\textit{SuperHyperConnectivities}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7606404).
\bibitem{HG102b} Henry Garrett, “\textit{Neutrosophic SuperHyperCycle}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7580018).
\bibitem{HG101b} Henry Garrett, “\textit{Extreme SuperHyperCycle}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7580018).
\bibitem{HG100b} Henry Garrett, “\textit{Extreme SuperHyperCycle}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7580018).
\bibitem{HG99b} Henry Garrett, “\textit{SuperHyperCycle}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7579929).
\bibitem{HG98b} Henry Garrett, “\textit{Neutrosophic SuperHyperGirth}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7563170).
\bibitem{HG97b} Henry Garrett, “\textit{Extreme SuperHyperGirth}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7563164).
\bibitem{HG96b} Henry Garrett, “\textit{SuperHyperGirth}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7563160).
\bibitem{HG95b} Henry Garrett, “\textit{Extreme SuperHyperGirth}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7563160).
\bibitem{HG94b} Henry Garrett, “\textit{Overlook On SuperHyperGirth}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7563160).
\bibitem{HG93b} Henry Garrett, “\textit{Neutrosophic SuperHyperMatching}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7557063).
\bibitem{HG92b} Henry Garrett, “\textit{Extreme SuperHyperMatching}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7557009).
\bibitem{HG91b} Henry Garrett, “\textit{Overlook On SuperHyperMatching}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7539484).
\bibitem{HG90b} Henry Garrett, “\textit{Neutrosophic Failed SuperHyperClique}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7523390).
\bibitem{HG89b} Henry Garrett, “\textit{Extreme Failed SuperHyperClique}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7523390).
\bibitem{HG88b} Henry Garrett, “\textit{Overlook On Failed SuperHyperClique}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7523390).
\bibitem{HG87b} Henry Garrett, “\textit{Extreme SuperHyperClique}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7574952).
\bibitem{HG86b} Henry Garrett, “\textit{Neutrosophic SuperHyperClique}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7574992).
\bibitem{HG85b} Henry Garrett, “\textit{Extreme SuperHyperClique}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7523357).
\bibitem{HG84b} Henry Garrett, “\textit{Overlook On SuperHyperClique}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7523357).
\bibitem{HG83b} Henry Garrett, “\textit{Neutrosophic Failed SuperHyperStable}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7504782).
\bibitem{HG82b} Henry Garrett, “\textit{Extreme Failed SuperHyperStable}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7504782).
\bibitem{HG81b} Henry Garrett, “\textit{Overlook On Failed SuperHyperStable}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7504782).
\bibitem{HG80b} Henry Garrett, “\textit{Neutrosophic SuperHyperStable}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7499395).
\bibitem{HG79b} Henry Garrett, “\textit{Extreme SuperHyperStable}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7499395).
\bibitem{HG78b} Henry Garrett, “\textit{Overlook On SuperHyperStable}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7499395).
\bibitem{HG77b} Henry Garrett, “\textit{Neutrosophic Failed SuperHyperForcing}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7497450).
\bibitem{HG76b} Henry Garrett, “\textit{Extreme Failed SuperHyperForcing}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7497450).
\bibitem{HG75b} Henry Garrett, “\textit{Neutrosophic SuperHyperForcing}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7494862).
\bibitem{HG74b} Henry Garrett, “\textit{Extreme SuperHyperForcing}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7494862).
\bibitem{HG73b} Henry Garrett, “\textit{Overlook On SuperHyperForcing}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7494862).
\bibitem{HG72b} Henry Garrett, “\textit{Neutrosophic SuperHyperAlliances}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7493845).
\bibitem{HG71b} Henry Garrett, “\textit{Extreme SuperHyperAlliances}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7493845).
\bibitem{HG70b} Henry Garrett, “\textit{Overlook On SuperHyperAlliances}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7493845).
\bibitem{HG69b} Henry Garrett, “\textit{SuperHyperMatching}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7539484).
\bibitem{HG68b} Henry Garrett, “\textit{Failed SuperHyperClique}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7523390).
\bibitem{HG67b} Henry Garrett, “\textit{SuperHyperClique}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7523357).
\bibitem{HG66b} Henry Garrett, “\textit{Failed SuperHyperStable}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7504782).
\bibitem{HG65b} Henry Garrett, “\textit{SuperHyperStable}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7499395).
\bibitem{HG64b} Henry Garrett, “\textit{Failed SuperHyperForcing}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7497450).
\bibitem{HG63b} Henry Garrett, “\textit{SuperHyperForcing}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7494862).
\bibitem{HG62b} Henry Garrett, “\textit{SuperHyperAlliances}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7493845).
\bibitem{HG61b} Henry Garrett, “\textit{SuperHyperGraphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7480110).
\bibitem{HG60b} Henry Garrett, “\textit{Neut. SuperHyperEdges}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.7378758).
\bibitem{HG32b} Henry Garrett, “\textit{Beyond Neutrosophic Graphs}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.6320305).
\bibitem{HG44b} Henry Garrett, “\textit{Neutrosophic Duality}”. Dr. Henry Garrett, 2023 (doi: 10.5281/zenodo.6677173).
\end{thebibliography}
\end{document}