“#183 Article” Henry Garrett, “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). @ResearchGate: https://www.researchgate.net/publication/369087468 @Scribd: https://www.scribd.com/document/- @ZENODO_ORG: https://zenodo.org/record/- @academia: https://www.academia.edu/- \documentclass[10pt,letterpaper]{article} \usepackage[top=0.85in,left=2.79in,footskip=0.79in,marginparwidth=2in]{geometry} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, tikz, color} \usepackage[bookmarksnumbered, colorlinks, plainpages]{hyperref} % use Unicode characters - try changing the option if you run into troubles with special characters (e.g. umlauts) \usepackage[utf8]{inputenc} % clean citations \usepackage{cite} % hyperref makes references clicky. use \url{www.example.com} or \href{www.example.com}{description} to add a clicky url \usepackage{nameref,hyperref} % line numbers \usepackage[right]{lineno} % improves typesetting in LaTeX \usepackage{microtype} \DisableLigatures[f]{encoding = *, family = * } % text layout - change as needed \raggedright \setlength{\parindent}{0.5cm} \textwidth 5.25in \textheight 8.79in % Remove % for double line spacing %\usepackage{setspace} %\doublespacing % use adjustwidth environment to exceed text width (see examples in text) \usepackage{changepage} % adjust caption style \usepackage[aboveskip=1pt,labelfont=bf,labelsep=period,singlelinecheck=off]{caption} % remove brackets from references \makeatletter \renewcommand{\@biblabel}[1]{\quad#1.} \makeatother % headrule, footrule and page numbers \usepackage{lastpage,fancyhdr,graphicx} \usepackage{epstopdf} \pagestyle{myheadings} \pagestyle{fancy} \fancyhf{} \rfoot{\thepage/\pageref{LastPage}} \renewcommand{\footrule}{\hrule height 2pt \vspace{2mm}} \fancyheadoffset[L]{2.25in} \fancyfootoffset[L]{2.25in} % use \textcolor{color}{text} for colored text (e.g. highlight to-do areas) \usepackage{color} % define custom colors (this one is for figure captions) \definecolor{Gray}{gray}{.25} % this is required to include graphics \usepackage{graphicx} % use if you want to put caption to the side of the figure - see example in text \usepackage{sidecap} \usepackage{leftidx} % use for have text wrap around figures \usepackage{wrapfig} \usepackage[pscoord]{eso-pic} \usepackage[fulladjust]{marginnote} \reversemarginpar \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \theoremstyle{observation} \newtheorem{observation}[theorem]{Observation} \theoremstyle{question} \newtheorem{question}[theorem]{Question} \theoremstyle{problem} \newtheorem{problem}[theorem]{Problem} \numberwithin{equation}{section} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \fancyhead[LE,RO]{Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA } \fancyfoot[LE,RO]{Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA } % document begins here \begin{document} \vspace*{0.35in} \linenumbers % title goes here: \begin{flushleft} {\Large \textbf\newline{ New Ideas On Super Solidarity By Hyper Soul Of Space In Cancer's Recognition With (Extreme) SuperHyperGraph } } \newline \newline Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA % authors go here: \end{flushleft} \section{ABSTRACT} In this scientific research, (Different Neutrosophic Types of Neutrosophic SuperHyperSpace). Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a Space 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-SuperHyperSpace if $S=(V,E)$ is a probability space where $V$ is a sample space and $E$ is a function space $E:V\rightarrow [0,1]$ such that $\sum_{ab\in V}E(ab)=1$; Neutrosophic re-SuperHyperSpace if $S=(V,E)$ is a probability space where $V$ is a sample space and $E$ is a function space $E:V\rightarrow [0,1]$ such that $\sum_{ab\in V}E(ab)=1$; and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$ Neutrosophic v-SuperHyperSpace if $S=(V,E)$ is a probability space where $V$ is a sample space and $E$ is a function space $E:V\rightarrow [0,1]$ such that $\sum_{ab\in V}E(ab)=1$; Neutrosophic rv-SuperHyperSpace if $S=(V,E)$ is a probability space where $V$ is a sample space and $E$ is a function space $E:V\rightarrow [0,1]$ such that $\sum_{ab\in V}E(ab)=1$; and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$ Neutrosophic SuperHyperSpace if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace. ((Neutrosophic) SuperHyperSpace). 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 SuperHyperSpace if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperSpace; a Neutrosophic SuperHyperSpace if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperSpace; an Extreme SuperHyperSpace SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperSpace; and the Extreme power is corresponded to its Extreme coefficient; a Neutrosophic SuperHyperSpace SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperSpace; and the Neutrosophic power is corresponded to its Neutrosophic coefficient; an Extreme V-SuperHyperSpace if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperSpace; a Neutrosophic V-SuperHyperSpace if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperSpace; an Extreme V-SuperHyperSpace SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperSpace; and the Extreme power is corresponded to its Extreme coefficient; a Neutrosophic SuperHyperSpace SuperHyperPolynomial if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperSpace; and the Neutrosophic power is corresponded to its Neutrosophic coefficient. In this scientific research, new setting is introduced for new SuperHyperNotions, namely, a SuperHyperSpace and Neutrosophic SuperHyperSpace. 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 $\delta-$SuperHyperSpace 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))|_{Neutrosophic}+\delta;$ and $ |S\cap N(s)|_{Neutrosophic} , x\in X\}$$ where the functions $T, I, F: X\rightarrow]^-0,1\leftidx{^+}{[}$ define respectively the a \textbf{truth-membership function}, an \textbf{indeterminacy-membership function}, and a \textbf{falsity-membership function} of the element $x\in X$ to the set $A$ with the condition $$\leftidx{^-}{0} \leq T_A(x)+I_A(x)+F_A(x)\leq 3^{+}.$$ The functions $T_A(x),I_A(x)$ and $F_A(x)$ are real standard or nonstandard subsets of $]^-0,1\leftidx{^+}{[}.$ \end{definition} \begin{definition}[Single Valued Neutrosophic Set](\textbf{Ref.}\cite{HG38},Definition 2.2,p.2).\\ Let $X$ be a space of points (objects) with generic elements in $X$ denoted by $x.$ A \textbf{single valued Neutrosophic set} $A$ (SVNS $A$) is characterized by truth-membership function $T_A(x),$ an indeterminacy-membership function $I_A(x),$ and a falsity-membership function $F_A(x).$ For each point $x$ in $X,$ $T_A(x),I_A(x),F_A(x)\in [0,1].$ A SVNS $A$ can be written as $$A = \{, x\in X\}.$$ \end{definition} \begin{definition} The \textbf{degree of truth-membership}, \textbf{indeterminacy-membership} and \textbf{falsity-membership of the subset} $X\subset A$ of the single valued Neutrosophic set $A = \{, x\in X\}$:$$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}.$$ \end{definition} \begin{definition} The \textbf{support} of $X\subset A$ of the single valued Neutrosophic set $A = \{, x\in X\}$: $$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\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}.$$ \end{definition} \begin{definition} The \textbf{support} of $X\subset A$ of the single valued Neutrosophic set $A = \{, x\in X\}$: $$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 SuperHyperSpace).\\ 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-SuperHyperSpace} if $S=(V,E)$ is a probability space where $V$ is a sample space and $E$ is a function space $E:V\rightarrow [0,1]$ such that $\sum_{ab\in V}E(ab)=1$; \item[$(ii)$] \textbf{Neutrosophic re-SuperHyperSpace} if $S=(V,E)$ is a probability space where $V$ is a sample space and $E$ is a function space $E:V\rightarrow [0,1]$ such that $\sum_{ab\in V}E(ab)=1$; and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$ \item[$(iii)$] \textbf{Neutrosophic v-SuperHyperSpace} if $S=(V,E)$ is a probability space where $V$ is a sample space and $E$ is a function space $E:V\rightarrow [0,1]$ such that $\sum_{ab\in V}E(ab)=1$; \item[$(iv)$] \textbf{Neutrosophic rv-SuperHyperSpace} if $S=(V,E)$ is a probability space where $V$ is a sample space and $E$ is a function space $E:V\rightarrow [0,1]$ such that $\sum_{ab\in V}E(ab)=1$; and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$ \item[$(v)$] \textbf{Neutrosophic SuperHyperSpace} if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace. \end{itemize} \end{definition} \begin{definition}((Neutrosophic) SuperHyperSpace).\\ 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 SuperHyperSpace} if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperSpace; \item[$(ii)$] a \textbf{Neutrosophic SuperHyperSpace} if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperSpace; \item[$(iii)$] an \textbf{Extreme SuperHyperSpace SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperSpace; and the Extreme power is corresponded to its Extreme coefficient; \item[$(iv)$] a \textbf{Neutrosophic SuperHyperSpace SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperSpace; and the Neutrosophic power is corresponded to its Neutrosophic coefficient; \item[$(v)$] an \textbf{Extreme V-SuperHyperSpace} if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperSpace; \item[$(vi)$] a \textbf{Neutrosophic V-SuperHyperSpace} if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperSpace; \item[$(vii)$] an \textbf{Extreme V-SuperHyperSpace SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperSpace; and the Extreme power is corresponded to its Extreme coefficient; \item[$(viii)$] a \textbf{Neutrosophic SuperHyperSpace SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperSpace, Neutrosophic re-SuperHyperSpace, Neutrosophic v-SuperHyperSpace, and Neutrosophic rv-SuperHyperSpace and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality consecutive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperSpace; and the Neutrosophic power is corresponded to its Neutrosophic coefficient. \end{itemize} \end{definition} \begin{definition}((Extreme/Neutrosophic)$\delta-$SuperHyperSpace).\\ Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Then \begin{itemize} \item[$(i)$] an \textbf{$\delta-$SuperHyperSpace} is a Neutrosophic kind of Neutrosophic SuperHyperSpace 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))|_{Neutrosophic}+\delta; \label{136EQN3} \\&& |S\cap N(s)|_{Neutrosophic} \text{l choose 2} \geq cr(G) \geq {(kl)}^3/64n^2$ by the Extreme Crossing Lemma, and again $l n(n-1) ≥ cr(G) ≥ {(k-n)}^3/64n^2$ by the Extreme Crossing Lemma, and $k> 1,$ in which case almost surely $\alpha(G)$ is equal to either $k^{*}-1$ or $k^{*}.$ \end{itemize} \end{corollary} \begin{proof} Assume an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability space. The latter is straightforward. \end{proof} \begin{definition}(Extreme Threshold).\\ Assume an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability space. Let $P$ be a monotone property of SuperHyperGraphs (one which is preserved when SuperHyperEdges are added). Then a \textbf{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 has $P.$ \end{itemize} \end{definition} \begin{definition}(Extreme Balanced).\\ Assume an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability space. Let $F$ be a fixed Extreme SuperHyperGraph. Then there is a threshold function for the property of containing a copy of $F$ as an Extreme SubSuperHyperGraph is called \textbf{Extreme Balanced}. \end{definition} \begin{theorem} Assume an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability space. Let $F$ be a nonempty balanced Extreme SubSuperHyperGraph with $k$ SuperHyperVertices and $l$ SuperHyperEdges. Then $n^{-k/l}$ is a threshold function for the property of containing $F$ as an Extreme SubSuperHyperGraph. \end{theorem} \begin{proof} Assume an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability space. The latter is straightforward. \end{proof} \begin{example}\label{136EXM1} Assume an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Extreme Figures in every Extreme items. \begin{itemize} \item On the Figure \eqref{136NSHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. $E_1$ and $E_3$ are some empty Extreme SuperHyperEdges but $E_2$ is a loop Extreme SuperHyperEdge and $E_4$ is an Extreme SuperHyperEdge. Thus in the terms of Extreme SuperHyperNeighbor, there's only one Extreme SuperHyperEdge, namely, $E_4.$ The Extreme SuperHyperVertex, $V_3$ is Extreme isolated means that there's no Extreme SuperHyperEdge has it as an Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given Extreme SuperHyperSpace. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}}=\{E_4\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}}=z. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}}=\{V_i\}_{i\neq3}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}}=z^3. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG1.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG1} \end{figure} \item On the Figure \eqref{136NSHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. $E_1,E_2$ and $E_3$ are some empty Extreme SuperHyperEdges but $E_4$ is an Extreme SuperHyperEdge. Thus in the terms of Extreme SuperHyperNeighbor, there's only one Extreme SuperHyperEdge, namely, $E_4.$ The Extreme SuperHyperVertex, $V_3$ is Extreme isolated means that there's no Extreme SuperHyperEdge has it as an Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given Extreme SuperHyperSpace. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}}=\{E_4\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}}=z. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}}=\{V_i\}_{i\neq3}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}}=z^3. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG2.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG2} \end{figure} \item On the Figure \eqref{136NSHG3}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}}=\{E_4\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}}=z. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}}=\{V_i\}_{i=1}^3. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}}=z^3. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG3.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG3} \end{figure} \item On the Figure \eqref{136NSHG4}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}}=\{E_1\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}}=z. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}}=\{V_3,V_4,H\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =z^3. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG4.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG4} \end{figure} \item On the Figure \eqref{136NSHG5}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}}=\{\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =0z^0. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}}=\{V_5\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =z. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG5.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG5} \end{figure} \item On the Figure \eqref{136NSHG6}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}} \\&& =\{E_1,E_2,E_{13}\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =az^3. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{V_1,V_3,V_{12}\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =bz^3. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG6.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG6} \end{figure} \item On the Figure \eqref{136NSHG7}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}}= \\&& \{E_{12},E_{13}\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =3z^2. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}}= \\&& \{V_1,V_3,V_{12}\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =3z^3. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG7.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG7} \end{figure} \item On the Figure \eqref{136NSHG8}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}}= \\&& \{E_4\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =4z. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}}= \\&& \{V_{12}\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =3z. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG8.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG8} \end{figure} \item On the Figure \eqref{136NSHG9}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}} \\&& =\{E_1,E_2,E_{13}\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =11z^3. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{V_1,V_3,V_{12}\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =11z^3. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG9.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG9} \end{figure} \item On the Figure \eqref{136NSHG10}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}}= \\&& \{E_4,E_5,E_6\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =z^3. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}}= \\&& =\{V_{12}\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =3z. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG10.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG10} \end{figure} \item On the Figure \eqref{136NSHG11}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}}= \\&& \{E_1,E_6\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =z^2. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}}=\{V_1,V_3,V_4\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =6z^3. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG11.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG11} \end{figure} \item On the Figure \eqref{136NSHG12}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}}=\{E_2\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =5z. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{V_1\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}}= \\&& =5z. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG12.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG12} \end{figure} \item On the Figure \eqref{136NSHG13}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}}= \\&& \{E_6,E_7,E_8\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =z^3. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}}=\{V_1,V_3,V_4\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =6z^3. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG13.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG13} \end{figure} \item On the Figure \eqref{136NSHG14}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}}=\{E_1\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}}=2z. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}}=\{V_1\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}}= z. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG14.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG14} \end{figure} \item On the Figure \eqref{136NSHG15}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}} \\&& =\{E_1\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}}=5z. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{V_1\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} =4z. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG15.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG15} \end{figure} \item On the Figure \eqref{136NSHG16}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}} \\&& =\{E_1\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}}=5z. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{V_2\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =z. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG16.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG16} \end{figure} \item On the Figure \eqref{136NSHG17}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}} \\&& =\{E_1\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}}=5z. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{V_2\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =z. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG17.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG17} \end{figure} \item On the Figure \eqref{136NSHG18}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}} \\&& =\{E_1\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}}=6z. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{V_2\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =2z. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG18.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG18} \end{figure} \item On the Figure \eqref{136NSHG19}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}} \\&& =\{E_1,E_3\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}}=12z^2. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{V_1,V_2\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =11z^2. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG19.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG19} \end{figure} \item On the Figure \eqref{136NSHG20}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}} \\&& =\{\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =0z^0. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{V_1\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =z. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{136NSHG20.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{136NSHG20} \end{figure} \item On the Figure \eqref{95NHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}} \\&& =\{E_2\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =z. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =0z^0. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{95NHG1.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{95NHG1} \end{figure} \item On the Figure \eqref{95NHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperSpace, is up. The Extreme Algorithm is Extremely straightforward. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}} \\&& =\{E_1\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =5z. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{V_6\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =z. \end{eqnarray*} \begin{figure} \includegraphics[width=100mm]{95NHG2.png} \caption{The Extreme SuperHyperGraphs Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM1} } \label{95NHG2} \end{figure} \end{itemize} \end{example} \section{The Extreme Departures on The Theoretical Results Toward Theoretical Motivations} The previous Extreme approach apply on the upcoming Extreme results on Extreme SuperHyperClasses. \begin{proposition} Assume a connected Extreme SuperHyperPath $ESHP:(V,E).$ Then \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}} \\&& =\{E_2\}_{i=1}^{|E_{NSHG}|}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =(|E_{NSHG}|-2)z. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{V^{EXTERNAL}_i\}_{V^{EXTERNAL}_i\in E_2}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =(|E_{NSHG}|-2)z^a. \end{eqnarray*} \end{proposition} \begin{proof} Let \begin{eqnarray*} && P: \\&& V^{EXTERNAL}_1,E_1, \\&&V^{EXTERNAL}_2,E_2, \\&&\ldots, \\&&V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}},E_{\frac{|E_{NSHG}|}{3}} \end{eqnarray*} \begin{eqnarray*} && P: \\&& E_1,V^{EXTERNAL}_1, \\&&E_2,V^{EXTERNAL}_2, \\&&\ldots, \\&&E_{\frac{|E_{NSHG}|}{3}},V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}} \end{eqnarray*} be a longest path taken from a connected Extreme SuperHyperPath $ESHP:(V,E).$ There's a new way to redefine as \begin{eqnarray*} && V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv \\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z \equiv \\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z. \end{eqnarray*} The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperSpace. The latter is straightforward. \end{proof} \begin{example}\label{136EXM18a} In the Figure \eqref{136NSHG18a}, the connected Extreme SuperHyperPath $ESHP:(V,E),$ is highlighted and featured. The Extreme SuperHyperSet, in the Extreme SuperHyperModel \eqref{136NSHG18a}, is the SuperHyperSpace. \begin{figure} \includegraphics[width=100mm]{136NSHG18.png} \caption{an Extreme SuperHyperPath Associated to the Notions of Extreme SuperHyperSpace in the Example \eqref{136EXM18a}} \label{136NSHG18a} \end{figure} \end{example} \begin{proposition} Assume a connected Extreme SuperHyperCycle $ESHC:(V,E).$ Then \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}} \\&& =\{E_1,E_3\}_{i=1}^{|E_{NSHG}|}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =(|E_{NSHG}|)z^2. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{V^{EXTERNAL}_i,V^{EXTERNAL}_j\}_{V^{EXTERNAL}_i\in E_1,V^{EXTERNAL}_j\in E_2}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =(|E_{NSHG}|)z^a. \end{eqnarray*} \end{proposition} \begin{proof} Let \begin{eqnarray*} && P: \\&& V^{EXTERNAL}_1,E_1, \\&&V^{EXTERNAL}_2,E_2, \\&&\ldots, \\&&V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}},E_{\frac{|E_{NSHG}|}{3}} \end{eqnarray*} \begin{eqnarray*} && P: \\&& E_1,V^{EXTERNAL}_1, \\&&E_2,V^{EXTERNAL}_2, \\&&\ldots, \\&&E_{\frac{|E_{NSHG}|}{3}},V^{EXTERNAL}_{\frac{|E_{NSHG}|}{3}} \end{eqnarray*} be a longest path taken from a connected Extreme SuperHyperCycle $ESHC:(V,E).$ There's a new way to redefine as \begin{eqnarray*} && V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv \\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z \equiv \\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z. \end{eqnarray*} The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperSpace. The latter is straightforward. \end{proof} \begin{example}\label{136EXM19a} In the Figure \eqref{136NSHG19a}, the connected Extreme SuperHyperCycle $NSHC:(V,E),$ is highlighted and featured. The obtained Extreme SuperHyperSet, in the Extreme SuperHyperModel \eqref{136NSHG19a}, is the Extreme SuperHyperSpace. \begin{figure} \includegraphics[width=100mm]{136NSHG19.png} \caption{an Extreme SuperHyperCycle Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM19a}} \label{136NSHG19a} \end{figure} \end{example} \begin{proposition} Assume a connected Extreme SuperHyperStar $ESHS:(V,E).$ Then \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}}=\{\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =0z^0. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{CENTER\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =z^0. \end{eqnarray*} \end{proposition} \begin{proof} Let \begin{eqnarray*} && P: \\&& V^{EXTERNAL}_1,E_1, \\&&CENTER,E_2 \end{eqnarray*} \begin{eqnarray*} && P: \\&& E_1,V^{EXTERNAL}_1, \\&&E_2,CENTER \end{eqnarray*} be a longest path taken a connected Extreme SuperHyperStar $ESHS:(V,E).$ There's a new way to redefine as \begin{eqnarray*} && V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv \\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z \equiv \\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z. \end{eqnarray*} The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperSpace. The latter is straightforward. \end{proof} \begin{example}\label{136EXM20a} In the Figure \eqref{136NSHG20a}, the connected Extreme SuperHyperStar $ESHS:(V,E),$ is highlighted and featured. The obtained Extreme SuperHyperSet, by the Algorithm in previous Extreme result, of the Extreme SuperHyperVertices of the connected Extreme SuperHyperStar $ESHS:(V,E),$ in the Extreme SuperHyperModel \eqref{136NSHG20a}, is the Extreme SuperHyperSpace. \begin{figure} \includegraphics[width=100mm]{136NSHG20.png} \caption{an Extreme SuperHyperStar Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM20a}} \label{136NSHG20a} \end{figure} \end{example} \begin{proposition} Assume a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Then \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}} \\&& =\{E_i,~E_i\in P_1\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =(|P^{i}_{NSHG}|\times|P^{j}_{NSHG}|)z^{|P^{\min}_{NSHG}|}. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{V^{EXTERNAL}_i,V^{EXTERNAL}_{V^{EXTERNAL}\in P_1\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =(|P^{i}_{NSHG}|\times|P^{j}_{NSHG}|)z^{|P^{\min}_{NSHG}|}. \end{eqnarray*} \end{proposition} \begin{proof} Let \begin{eqnarray*} && P: \\&& V^{EXTERNAL}_1,E_1, \\&&V^{EXTERNAL}_2,E_2, \\&&\ldots, \\&&V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|}. \end{eqnarray*} \begin{eqnarray*} && P: \\&& E_1,V^{EXTERNAL}_1, \\&&E_2,V^{EXTERNAL}_2, \\&&\ldots, \\&&E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|} \end{eqnarray*} is a longest path taken from a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ There's a new way to redefine as \begin{eqnarray*} && V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv \\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z \equiv \\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z. \end{eqnarray*} The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperSpace. The latter is straightforward. Then there's no at least one SuperHyperSpace. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperSpace 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 SuperHyperSpace taken from a connected Extreme SuperHyperBipartite $ESHB:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-of-SuperHyperPart SuperHyperEdges are attained in any solution \begin{eqnarray*} && P: \\&& V^{EXTERNAL}_1,E_1, \\&&V^{EXTERNAL}_2,E_2 \end{eqnarray*} The latter is straightforward. \end{proof} \begin{example}\label{136EXM21a} In the Extreme Figure \eqref{136NSHG21a}, the connected Extreme SuperHyperBipartite $ESHB:(V,E),$ is Extreme highlighted and Extreme featured. The obtained Extreme SuperHyperSet, by the Extreme Algorithm in previous Extreme result, of the Extreme SuperHyperVertices of the connected Extreme SuperHyperBipartite $ESHB:(V,E),$ in the Extreme SuperHyperModel \eqref{136NSHG21a}, is the Extreme SuperHyperSpace. \begin{figure} \includegraphics[width=100mm]{136NSHG21.png} \caption{Extreme SuperHyperBipartite Extreme Associated to the Extreme Notions of Extreme SuperHyperSpace in the Example \eqref{136EXM21a}} \label{136NSHG21a} \end{figure} \end{example} \begin{proposition} Assume a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$ Then \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}} \\&& =\{E_i,~E_i\in P_1\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =(|P^{i}_{NSHG}|\times|P^{j}_{NSHG}|)z^{|P^{\min}_{NSHG}|}. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& =\{V^{EXTERNAL}_i,V^{EXTERNAL}_{V^{EXTERNAL}\in P_1\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =(|P^{i}_{NSHG}|\times|P^{j}_{NSHG}|)z^{|P^{\min}_{NSHG}|}. \end{eqnarray*} \end{proposition} \begin{proof} Let \begin{eqnarray*} && P: \\&& V^{EXTERNAL}_1,E_1, \\&&V^{EXTERNAL}_2,E_2, \\&&\ldots, \\&&V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|}. \end{eqnarray*} \begin{eqnarray*} && P: \\&& E_1,V^{EXTERNAL}_1, \\&&E_2,V^{EXTERNAL}_2, \\&&\ldots, \\&&E_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|},V^{EXTERNAL}_{|P_i|=\min_{P_j\in E_{NSHG}}|P_j|} \end{eqnarray*} is a longest SuperHyperSpace taken from a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$ There's a new way to redefine as \begin{eqnarray*} && V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv \\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z \equiv \\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z. \end{eqnarray*} The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperSpace. The latter is straightforward. Then there's no at least one SuperHyperSpace. Thus the notion of quasi may be up but the SuperHyperNotions based on SuperHyperSpace could be applied. There are only $z'$ SuperHyperParts. Thus every SuperHyperPart could have one SuperHyperVertex as the representative in the \begin{eqnarray*} && P: \\&& V^{EXTERNAL}_1,E_1, \\&&V^{EXTERNAL}_2,E_2 \end{eqnarray*} is a longest path taken from a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$ Thus only some SuperHyperVertices and only minimum-Extreme-of-SuperHyperPart SuperHyperEdges are attained in any solution \begin{eqnarray*} && P: \\&& V^{EXTERNAL}_1,E_1, \\&&V^{EXTERNAL}_2,E_2 \end{eqnarray*} is a longest path taken from a connected Extreme SuperHyperMultipartite $ESHM:(V,E).$ The latter is straightforward. \end{proof} \begin{example}\label{136EXM22a} In the Figure \eqref{136NSHG22a}, the connected Extreme SuperHyperMultipartite $ESHM:(V,E),$ is highlighted and Extreme featured. The obtained Extreme SuperHyperSet, by the Algorithm in previous Extreme result, of the Extreme SuperHyperVertices of the connected Extreme SuperHyperMultipartite $ESHM:(V,E),$ in the Extreme SuperHyperModel \eqref{136NSHG22a}, is the Extreme SuperHyperSpace. \begin{figure} \includegraphics[width=100mm]{136NSHG22.png} \caption{an Extreme SuperHyperMultipartite Associated to the Notions of Extreme SuperHyperSpace in the Example \eqref{136EXM22a}} \label{136NSHG22a} \end{figure} \end{example} \begin{proposition} Assume a connected Extreme SuperHyperWheel $ESHW:(V,E).$ Then, \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme Space}}=\{E^{*}_i,E_1,E_3\}_{i=1}^{|E^{*}_{NSHG}|}. \\&& \mathcal{C}(NSHG)_{\text{Extreme Space SuperHyperPolynomial}} \\&& =|E_{NSHG}|z^{|E^{*}_{NSHG}|+2}. \\&& \mathcal{C}(NSHG)_{\text{Extreme V-Space}} \\&& = \{V^{EXTERNAL}_1,V^{EXTERNAL}_3,CENTER\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme V-Space SuperHyperPolynomial}}} \\&& =||E_{NSHG}|z^3. \end{eqnarray*} \end{proposition} \begin{proof} Let \begin{eqnarray*} && P: \\&& V^{EXTERNAL}_1,E^{*}_1, \\&&CENTER,E^{*}_2 \end{eqnarray*} \begin{eqnarray*} && P: \\&& E^{*}_1,V^{EXTERNAL}_1, \\&&E^{*}_2,CENTER \end{eqnarray*} is a longest SuperHyperSpace taken from a connected Extreme SuperHyperWheel $ESHW:(V,E).$ There's a new way to redefine as \begin{eqnarray*} && V^{EXTERNAL}_i\sim V^{EXTERNAL}_j \equiv \\&& \exists! E_z\in E_{ESHG:(V,E)},~ V^{EXTERNAL}_i,V^{EXTERNAL}_j \in E_z \equiv \\&& \exists! E_z\in E_{ESHG:(V,E)},~ \{V^{EXTERNAL}_i,V^{EXTERNAL}_j\} \subseteq E_z. \end{eqnarray*} The term ``EXTERNAL'' implies $|N(V^{EXTERNAL}_i)|\geq |N(V_j)|$ where $V_j$ is corresponded to $V^{EXTERNAL}_i$ in the literatures of SuperHyperSpace. The latter is straightforward. Then there's at least one SuperHyperSpace. Thus the notion of quasi isn't up and the SuperHyperNotions based on SuperHyperSpace could be applied. The unique embedded SuperHyperSpace proposes some longest SuperHyperSpace excerpt from some representatives. The latter is straightforward. \end{proof} \begin{example}\label{136EXM23a} In the Extreme Figure \eqref{136NSHG23a}, the connected Extreme SuperHyperWheel $NSHW:(V,E),$ is Extreme highlighted and featured. The obtained Extreme SuperHyperSet, by the Algorithm in previous result, of the Extreme SuperHyperVertices of the connected Extreme SuperHyperWheel $ESHW:(V,E),$ in the Extreme SuperHyperModel \eqref{136NSHG23a}, is the Extreme SuperHyperSpace. \begin{figure} \includegraphics[width=100mm]{136NSHG23.png} \caption{an Extreme SuperHyperWheel Extreme Associated to the Extreme Notions of Extreme SuperHyperSpace in the Extreme Example \eqref{136EXM23a}} \label{136NSHG23a} \end{figure} \end{example} \section{The Surveys of Mathematical Sets On The Results But As The Initial Motivation} For the SuperHyperSpace, Extreme SuperHyperSpace, and the Extreme SuperHyperSpace, some general results are introduced. \begin{remark} Let remind that the Extreme SuperHyperSpace is ``redefined'' on the positions of the alphabets. \end{remark} \begin{corollary} Assume Extreme SuperHyperSpace. Then \begin{eqnarray*} && Extreme ~SuperHyperSpace=\\&&\{the SuperHyperSpace of the SuperHyperVertices ~|~\\&&\max|SuperHyperOffensive \\&&SuperHyperSpace \\&& |_{Extreme cardinality amid those SuperHyperSpace.}\} \end{eqnarray*} plus one Extreme SuperHypeNeighbor to one. Where $\sigma_i$ is the unary operation on the SuperHyperVertices of the SuperHyperGraph to assign the determinacy, the indeterminacy and the neutrality, for $i=1,2,3,$ respectively. \end{corollary} \begin{corollary} Assume an Extreme SuperHyperGraph on the same identical letter of the alphabet. Then the notion of Extreme SuperHyperSpace and SuperHyperSpace coincide. \end{corollary} \begin{corollary} Assume an Extreme SuperHyperGraph on the same identical letter of the alphabet. Then a consecutive sequence of the SuperHyperVertices is an Extreme SuperHyperSpace if and only if it's a SuperHyperSpace. \end{corollary} \begin{corollary} Assume an Extreme SuperHyperGraph on the same identical letter of the alphabet. Then a consecutive sequence of the SuperHyperVertices is a strongest SuperHyperSpace if and only if it's a longest SuperHyperSpace. \end{corollary} \begin{corollary} Assume SuperHyperClasses of an Extreme SuperHyperGraph on the same identical letter of the alphabet. Then its Extreme SuperHyperSpace is its SuperHyperSpace and reversely. \end{corollary} \begin{corollary} Assume an Extreme SuperHyperPath(-/SuperHyperSpace, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel) on the same identical letter of the alphabet. Then its Extreme SuperHyperSpace is its SuperHyperSpace and reversely. \end{corollary} \begin{corollary} Assume an Extreme SuperHyperGraph. Then its Extreme SuperHyperSpace isn't well-defined if and only if its SuperHyperSpace isn't well-defined. \end{corollary} \begin{corollary} Assume SuperHyperClasses of an Extreme SuperHyperGraph. Then its Extreme SuperHyperSpace isn't well-defined if and only if its SuperHyperSpace isn't well-defined. \end{corollary} \begin{corollary} Assume an Extreme SuperHyperPath(-/SuperHyperSpace, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). Then its Extreme SuperHyperSpace isn't well-defined if and only if its SuperHyperSpace isn't well-defined. \end{corollary} \begin{corollary} Assume an Extreme SuperHyperGraph. Then its Extreme SuperHyperSpace is well-defined if and only if its SuperHyperSpace is well-defined. \end{corollary} \begin{corollary} Assume SuperHyperClasses of an Extreme SuperHyperGraph. Then its Extreme SuperHyperSpace is well-defined if and only if its SuperHyperSpace is well-defined. \end{corollary} \begin{corollary} Assume an Extreme SuperHyperPath(-/SuperHyperSpace, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). Then its Extreme SuperHyperSpace is well-defined if and only if its SuperHyperSpace is well-defined. \end{corollary} % \begin{proposition} Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph. Then $V$ is \begin{itemize} \item[$(i):$] the dual SuperHyperDefensive SuperHyperSpace; \item[$(ii):$] the strong dual SuperHyperDefensive SuperHyperSpace; \item[$(iii):$] the connected dual SuperHyperDefensive SuperHyperSpace; \item[$(iv):$] the $\delta$-dual SuperHyperDefensive SuperHyperSpace; \item[$(v):$] the strong $\delta$-dual SuperHyperDefensive SuperHyperSpace; \item[$(vi):$] the connected $\delta$-dual SuperHyperDefensive SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $NTG:(V,E,\sigma,\mu)$ be an Extreme SuperHyperGraph. Then $\emptyset$ is \begin{itemize} \item[$(i):$] the SuperHyperDefensive SuperHyperSpace; \item[$(ii):$] the strong SuperHyperDefensive SuperHyperSpace; \item[$(iii):$] the connected defensive SuperHyperDefensive SuperHyperSpace; \item[$(iv):$] the $\delta$-SuperHyperDefensive SuperHyperSpace; \item[$(v):$] the strong $\delta$-SuperHyperDefensive SuperHyperSpace; \item[$(vi):$] the connected $\delta$-SuperHyperDefensive SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph. Then an independent SuperHyperSet is \begin{itemize} \item[$(i):$] the SuperHyperDefensive SuperHyperSpace; \item[$(ii):$] the strong SuperHyperDefensive SuperHyperSpace; \item[$(iii):$] the connected SuperHyperDefensive SuperHyperSpace; \item[$(iv):$] the $\delta$-SuperHyperDefensive SuperHyperSpace; \item[$(v):$] the strong $\delta$-SuperHyperDefensive SuperHyperSpace; \item[$(vi):$] the connected $\delta$-SuperHyperDefensive SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be an Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyperSpace/SuperHyperPath. Then $V$ is a maximal \begin{itemize} \item[$(i):$] SuperHyperDefensive SuperHyperSpace; \item[$(ii):$] strong SuperHyperDefensive SuperHyperSpace; \item[$(iii):$] connected SuperHyperDefensive SuperHyperSpace; \item[$(iv):$] $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyperSpace; \item[$(v):$] strong $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyperSpace; \item[$(vi):$] connected $\mathcal{O}(ESHG)$-SuperHyperDefensive SuperHyperSpace; \end{itemize} Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide. \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph which is a SuperHyperUniform SuperHyperWheel. Then $V$ is a maximal \begin{itemize} \item[$(i):$] dual SuperHyperDefensive SuperHyperSpace; \item[$(ii):$] strong dual SuperHyperDefensive SuperHyperSpace; \item[$(iii):$] connected dual SuperHyperDefensive SuperHyperSpace; \item[$(iv):$] $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyperSpace; \item[$(v):$] strong $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyperSpace; \item[$(vi):$] connected $\mathcal{O}(ESHG)$-dual SuperHyperDefensive SuperHyperSpace; \end{itemize} Where the exterior SuperHyperVertices and the interior SuperHyperVertices coincide. \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be an Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyperSpace/SuperHyperPath. Then the number of \begin{itemize} \item[$(i):$] the SuperHyperSpace; \item[$(ii):$] the SuperHyperSpace; \item[$(iii):$] the connected SuperHyperSpace; \item[$(iv):$] the $\mathcal{O}(ESHG)$-SuperHyperSpace; \item[$(v):$] the strong $\mathcal{O}(ESHG)$-SuperHyperSpace; \item[$(vi):$] the connected $\mathcal{O}(ESHG)$-SuperHyperSpace. \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 an Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyperWheel. Then the number of \begin{itemize} \item[$(i):$] the dual SuperHyperSpace; \item[$(ii):$] the dual SuperHyperSpace; \item[$(iii):$] the dual connected SuperHyperSpace; \item[$(iv):$] the dual $\mathcal{O}(ESHG)$-SuperHyperSpace; \item[$(v):$] the strong dual $\mathcal{O}(ESHG)$-SuperHyperSpace; \item[$(vi):$] the connected dual $\mathcal{O}(ESHG)$-SuperHyperSpace. \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 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 SuperHyperSpace; \item[$(ii):$] strong dual SuperHyperDefensive SuperHyperSpace; \item[$(iii):$] connected dual SuperHyperDefensive SuperHyperSpace; \item[$(iv):$] $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperSpace; \item[$(v):$] strong $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperSpace; \item[$(vi):$] connected $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be 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 SuperHyperSpace; \item[$(ii):$] strong SuperHyperDefensive SuperHyperSpace; \item[$(iii):$] connected SuperHyperDefensive SuperHyperSpace; \item[$(iv):$] $\delta$-SuperHyperDefensive SuperHyperSpace; \item[$(v):$] strong $\delta$-SuperHyperDefensive SuperHyperSpace; \item[$(vi):$] connected $\delta$-SuperHyperDefensive SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be an Extreme SuperHyperUniform SuperHyperGraph which is a SuperHyperStar/SuperHyperComplete SuperHyperBipartite/SuperHyperComplete SuperHyperMultipartite. Then Then the number of \begin{itemize} \item[$(i):$] dual SuperHyperDefensive SuperHyperSpace; \item[$(ii):$] strong dual SuperHyperDefensive SuperHyperSpace; \item[$(iii):$] connected dual SuperHyperDefensive SuperHyperSpace; \item[$(iv):$] $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperSpace; \item[$(v):$] strong $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperSpace; \item[$(vi):$] connected $\frac{\mathcal{O}(ESHG)}{2}+1$-dual SuperHyperDefensive SuperHyperSpace. \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 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 SuperHyperSpace; \item[$(ii):$] strong SuperHyperDefensive SuperHyperSpace; \item[$(iii):$] connected SuperHyperDefensive SuperHyperSpace; \item[$(iv):$] SuperHyperSpace; \item[$(v):$] strong 1-SuperHyperDefensive SuperHyperSpace; \item[$(vi):$] connected 1-SuperHyperDefensive SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph. Then the number is at most $\mathcal{O}(ESHG)$ and the Extreme number is at most $\mathcal{O}_n(ESHG).$ \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph which is SuperHyperComplete. The number is $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and the Extreme number is $\min\Sigma_{v\in\{v_1,v_2,\cdots,v_t\}_{t>\frac{\mathcal{O}(ESHG:(V,E))}{2}}\subseteq V}\sigma(v),$ in the setting of dual \begin{itemize} \item[$(i):$] SuperHyperDefensive SuperHyperSpace; \item[$(ii):$] strong SuperHyperDefensive SuperHyperSpace; \item[$(iii):$] connected SuperHyperDefensive SuperHyperSpace; \item[$(iv):$] $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperSpace; \item[$(v):$] strong $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperSpace; \item[$(vi):$] connected $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph which is $\emptyset.$ The number is $0$ and the Extreme number is $0,$ for an independent SuperHyperSet in the setting of dual \begin{itemize} \item[$(i):$] SuperHyperDefensive SuperHyperSpace; \item[$(ii):$] strong SuperHyperDefensive SuperHyperSpace; \item[$(iii):$] connected SuperHyperDefensive SuperHyperSpace; \item[$(iv):$] $0$-SuperHyperDefensive SuperHyperSpace; \item[$(v):$] strong $0$-SuperHyperDefensive SuperHyperSpace; \item[$(vi):$] connected $0$-SuperHyperDefensive SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph which is SuperHyperComplete. Then there's no independent SuperHyperSet. \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph which is SuperHyperSpace/SuperHyperPath/SuperHyperWheel. The number is $\mathcal{O}(ESHG:(V,E))$ and the Extreme number is $\mathcal{O}_n(ESHG:(V,E)),$ in the setting of a dual \begin{itemize} \item[$(i):$] SuperHyperDefensive SuperHyperSpace; \item[$(ii):$] strong SuperHyperDefensive SuperHyperSpace; \item[$(iii):$] connected SuperHyperDefensive SuperHyperSpace; \item[$(iv):$] $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyperSpace; \item[$(v):$] strong $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyperSpace; \item[$(vi):$] connected $\mathcal{O}(ESHG:(V,E))$-SuperHyperDefensive SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be an Extreme SuperHyperGraph which is SuperHyperStar/complete SuperHyperBipartite/complete SuperHyperMultiPartite. The number is $\frac{\mathcal{O}(ESHG:(V,E))}{2}+1$ and the Extreme number is $\min\Sigma_{v\in\{v_1,v_2,\cdots,v_t\}_{t>\frac{\mathcal{O}(ESHG:(V,E))}{2}}\subseteq V}\sigma(v),$ in the setting of a dual \begin{itemize} \item[$(i):$] SuperHyperDefensive SuperHyperSpace; \item[$(ii):$] strong SuperHyperDefensive SuperHyperSpace; \item[$(iii):$] connected SuperHyperDefensive SuperHyperSpace; \item[$(iv):$] $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperSpace; \item[$(v):$] strong $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperSpace; \item[$(vi):$] connected $(\frac{\mathcal{O}(ESHG:(V,E))}{2}+1)$-SuperHyperDefensive SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $\mathcal{NSHF}:(V,E)$ be a SuperHyperFamily of the $ESHGs:(V,E)$ Extreme SuperHyperGraphs which are from one-type SuperHyperClass which the result is obtained for the individuals. Then the results also hold for the SuperHyperFamily $\mathcal{NSHF}:(V,E)$ of these specific SuperHyperClasses of the Extreme SuperHyperGraphs. \end{proposition} % \begin{proposition} Let $ESHG:(V,E)$ be a strong Extreme SuperHyperGraph. If $S$ is a dual SuperHyperDefensive SuperHyperSpace, then $\forall v\in V\setminus S,~\exists x\in S$ such that \begin{itemize} \item[$(i)$] $v\in N_s(x);$ \item[$(ii)$] $vx\in E.$ \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be a strong Extreme SuperHyperGraph. If $S$ is a dual SuperHyperDefensive SuperHyperSpace, then \begin{itemize} \item[$(i)$] $S$ is SuperHyperSpace set; \item[$(ii)$] there's $S\subseteq S'$ such that $|S'|$ is SuperHyperChromatic number. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be a strong Extreme SuperHyperGraph. Then \begin{itemize} \item[$(i)$] $\Gamma\leq\mathcal{O};$ \item[$(ii)$] $\Gamma_s\leq\mathcal{O}_n.$ \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be a strong Extreme SuperHyperGraph which is connected. Then \begin{itemize} \item[$(i)$] $\Gamma\leq\mathcal{O}-1;$ \item[$(ii)$] $\Gamma_s\leq\mathcal{O}_n-\Sigma_{i=1}^{3}\sigma_i(x).$ \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be an odd SuperHyperPath. Then \begin{itemize} \item[$(i)$] the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n-1}\}$ is a dual SuperHyperDefensive SuperHyperSpace; \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 SuperHyperSpace. \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 SuperHyperSpace; \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 SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be an even SuperHyperSpace. Then \begin{itemize} \item[$(i)$] the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n}\}$ is a dual SuperHyperDefensive SuperHyperSpace; \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 SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be an odd SuperHyperSpace. Then \begin{itemize} \item[$(i)$] the SuperHyperSet $S=\{v_2,v_4,\cdots,v_{n-1}\}$ is a dual SuperHyperDefensive SuperHyperSpace; \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 SuperHyperSpace. \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 SuperHyperSpace; \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 SuperHyperSpace. \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 SuperHyperSpace; \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 SuperHyperSpace. \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 SuperHyperSpace; \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 SuperHyperSpace. \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 SuperHyperSpace; \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 SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $\mathcal{NSHF}:(V,E)$ be a $m$-SuperHyperFamily of Extreme SuperHyperStars with common Extreme SuperHyperVertex SuperHyperSet. Then \begin{itemize} \item[$(i)$] the SuperHyperSet $S=\{c_1,c_2,\cdots,c_m\}$ is a dual SuperHyperDefensive SuperHyperSpace 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 SuperHyperSpace for $\mathcal{NSHF}:(V,E).$ \end{itemize} \end{proposition} \begin{proposition} Let $\mathcal{NSHF}:(V,E)$ be an $m$-SuperHyperFamily of odd SuperHyperComplete SuperHyperGraphs with common Extreme SuperHyperVertex SuperHyperSet. Then \begin{itemize} \item[$(i)$] the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor+1}$ is a dual maximal SuperHyperDefensive SuperHyperSpace 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 SuperHyperSpace for $\mathcal{NSHF}:(V,E).$ \end{itemize} \end{proposition} \begin{proposition} Let $\mathcal{NSHF}:(V,E)$ be a $m$-SuperHyperFamily of even SuperHyperComplete SuperHyperGraphs with common Extreme SuperHyperVertex SuperHyperSet. Then \begin{itemize} \item[$(i)$] the SuperHyperSet $S=\{v_i\}_{i=1}^{\lfloor\frac{n}{2}\rfloor}$ is a dual SuperHyperDefensive SuperHyperSpace 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 SuperHyperSpace for $\mathcal{NSHF}:(V,E).$ \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be a strong Extreme SuperHyperGraph. Then following statements hold; \begin{itemize} \item[$(i)$] if $s\geq t$ and a SuperHyperSet $S$ of SuperHyperVertices is an t-SuperHyperDefensive SuperHyperSpace, then $S$ is an s-SuperHyperDefensive SuperHyperSpace; \item[$(ii)$] if $s\leq t$ and a SuperHyperSet $S$ of SuperHyperVertices is a dual t-SuperHyperDefensive SuperHyperSpace, then $S$ is a dual s-SuperHyperDefensive SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be a strong Extreme SuperHyperGraph. Then following statements hold; \begin{itemize} \item[$(i)$] if $s\geq t+2$ and a SuperHyperSet $S$ of SuperHyperVertices is an t-SuperHyperDefensive SuperHyperSpace, then $S$ is an s-SuperHyperPowerful SuperHyperSpace; \item[$(ii)$] if $s\leq t$ and a SuperHyperSet $S$ of SuperHyperVertices is a dual t-SuperHyperDefensive SuperHyperSpace, then $S$ is a dual s-SuperHyperPowerful SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ be a[an] [V-]SuperHyperUniform-strong-Extreme SuperHyperGraph. Then following statements hold; \begin{itemize} \item[$(i)$] if $\forall a\in S,~|N_s(a)\cap S| \lfloor \frac{r}{2}\rfloor+1,$ then $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive SuperHyperSpace; \item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is an V-SuperHyperDefensive SuperHyperSpace; \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 SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Extreme SuperHyperGraph. Then following statements hold; \begin{itemize} \item[$(i)$] $\forall a\in S,~|N_s(a)\cap S| \lfloor \frac{r}{2}\rfloor+1$ if $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive SuperHyperSpace; \item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an V-SuperHyperDefensive SuperHyperSpace; \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 SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Extreme SuperHyperGraph which is a SuperHyperComplete. Then following statements hold; \begin{itemize} \item[$(i)$] $\forall a\in S,~|N_s(a)\cap S| \lfloor \frac{\mathcal{O}-1}{2}\rfloor+1$ if $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive SuperHyperSpace; \item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an $(\mathcal{O}-1)$-SuperHyperDefensive SuperHyperSpace; \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 SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Extreme SuperHyperGraph which is a SuperHyperComplete. Then following statements hold; \begin{itemize} \item[$(i)$] if $\forall a\in S,~|N_s(a)\cap S| \lfloor \frac{\mathcal{O}-1}{2}\rfloor+1,$ then $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive SuperHyperSpace; \item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is $(\mathcal{O}-1)$-SuperHyperDefensive SuperHyperSpace; \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 SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Extreme SuperHyperGraph which is SuperHyperSpace. Then following statements hold; \begin{itemize} \item[$(i)$] $\forall a\in S,~|N_s(a)\cap S| 2$ if $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive SuperHyperSpace; \item[$(iii)$] $\forall a\in S,~|N_s(a)\cap V\setminus S|=0$ if $ESHG:(V,E)$ is an 2-SuperHyperDefensive SuperHyperSpace; \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 SuperHyperSpace. \end{itemize} \end{proposition} \begin{proposition} Let $ESHG:(V,E)$ is a[an] [V-]SuperHyperUniform-strong-Extreme SuperHyperGraph which is SuperHyperSpace. Then following statements hold; \begin{itemize} \item[$(i)$] if $\forall a\in S,~|N_s(a)\cap S| 2,$ then $ESHG:(V,E)$ is a dual 2-SuperHyperDefensive SuperHyperSpace; \item[$(iii)$] if $\forall a\in S,~|N_s(a)\cap V\setminus S|=0,$ then $ESHG:(V,E)$ is an 2-SuperHyperDefensive SuperHyperSpace; \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 SuperHyperSpace. \end{itemize} \end{proposition} \section{Extreme Applications in Cancer's Extreme Recognition} The cancer is the Extreme disease but the Extreme model is going to figure out what's going on this Extreme phenomenon. The special Extreme case of this Extreme disease is considered and as the consequences of the model, some parameters are used. The cells are under attack of this disease but the moves of the cancer in the special region are the matter of mind. The Extreme recognition of the cancer could help to find some Extreme treatments for this Extreme disease. \\ In the following, some Extreme steps are Extreme devised on this disease. \begin{description} \item[Step 1. (Extreme Definition)] The Extreme recognition of the cancer in the long-term Extreme function. \item[Step 2. (Extreme Issue)] The specific region has been assigned by the Extreme model [it's called Extreme SuperHyperGraph] and the long Extreme cycle of the move from the cancer is identified by this research. Sometimes the move of the cancer hasn't be easily identified since there are some determinacy, indeterminacy and neutrality about the moves and the effects of the cancer on that region; this event leads us to choose another model [it's said to be Extreme SuperHyperGraph] to have convenient perception on what's happened and what's done. \item[Step 3. (Extreme Model)] There are some specific Extreme models, which are well-known and they've got the names, and some general Extreme models. The moves and the Extreme traces of the cancer on the complex tracks and between complicated groups of cells could be fantasized by an Extreme SuperHyperPath(-/SuperHyperSpace, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). The aim is to find either the Extreme SuperHyperSpace or the Extreme SuperHyperSpace in those Extreme Extreme SuperHyperModels. \section{Case 1: The Initial Extreme Steps Toward Extreme SuperHyperBipartite as Extreme SuperHyperModel} \item[Step 4. (Extreme Solution)] In the Extreme Figure \eqref{136NSHGaa21aa}, the Extreme SuperHyperBipartite is Extreme highlighted and Extreme featured. \begin{figure} \includegraphics[width=100mm]{136NSHG21.png} \caption{an Extreme SuperHyperBipartite Associated to the Notions of Extreme SuperHyperSpace} \label{136NSHGaa21aa} \end{figure} \\ By using the Extreme Figure \eqref{136NSHGaa21aa} and the Table \eqref{136TBLaa21aa}, the Extreme SuperHyperBipartite is obtained. \\ The obtained Extreme SuperHyperSet, by the Extreme Algorithm in previous Extreme result, of the Extreme SuperHyperVertices of the connected Extreme SuperHyperBipartite $ESHB:(V,E),$ in the Extreme SuperHyperModel \eqref{136NSHGaa21aa}, is the Extreme SuperHyperSpace. \begin{table} \centering \caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Extreme SuperHyperBipartite} \begin{tabular}[t]{c|c} \hline The Values of The Vertices & The Number of Position in Alphabet\\ \hline The Values of The SuperVertices&The maximum Values of Its Vertices\\ \hline The Values of The Edges&The maximum Values of Its Vertices\\ \hline The Values of The HyperEdges&The maximum Values of Its Vertices\\ \hline The Values of The SuperHyperEdges&The maximum Values of Its Endpoints \\ \hline \end{tabular} \label{136TBLaa21aa} \end{table} \section{Case 2: The Increasing Extreme Steps Toward Extreme SuperHyperMultipartite as Extreme SuperHyperModel} \item[Step 4. (Extreme Solution)] In the Extreme Figure \eqref{136NSHGaa22aa}, the Extreme SuperHyperMultipartite is Extreme highlighted and Extreme featured. \begin{figure} \includegraphics[width=100mm]{136NSHG22.png} \caption{an Extreme SuperHyperMultipartite Associated to the Notions of Extreme SuperHyperSpace} \label{136NSHGaa22aa} \end{figure} \\ By using the Extreme Figure \eqref{136NSHGaa22aa} and the Table \eqref{136TBLaa22aa}, the Extreme SuperHyperMultipartite is obtained. \\ The obtained Extreme SuperHyperSet, by the Extreme Algorithm in previous result, of the Extreme SuperHyperVertices of the connected Extreme SuperHyperMultipartite $ESHM:(V,E),$ in the Extreme SuperHyperModel \eqref{136NSHGaa22aa}, is the Extreme SuperHyperSpace. \begin{table} \centering \caption{The Values of Vertices, SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Extreme SuperHyperMultipartite} \begin{tabular}[t]{c|c} \hline The Values of The Vertices & The Number of Position in Alphabet\\ \hline The Values of The SuperVertices&The maximum Values of Its Vertices\\ \hline The Values of The Edges&The maximum Values of Its Vertices\\ \hline The Values of The HyperEdges&The maximum Values of Its Vertices\\ \hline The Values of The SuperHyperEdges&The maximum Values of Its Endpoints \\ \hline \end{tabular} \label{136TBLaa22aa} \end{table} \end{description} \section{Wondering Open Problems But As The Directions To Forming The Motivations} In what follows, some ``problems'' and some ``questions'' are proposed. \\ The SuperHyperSpace and the Extreme SuperHyperSpace 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 SuperHyperSpace and the Extreme SuperHyperSpace? \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 SuperHyperSpace and the Extreme SuperHyperSpace? \end{question} \begin{problem} The SuperHyperSpace and the Extreme SuperHyperSpace do a SuperHyperModel for the Cancer's recognitions and they're based on SuperHyperSpace, are there else? \end{problem} \begin{problem} Which the fundamental SuperHyperNumbers are related to these SuperHyperNumbers types-results? \end{problem} \begin{problem} What's the independent research based on Cancer's recognitions concerning the multiple types of SuperHyperNotions? \end{problem} \section{Conclusion and Closing Remarks} In this section, concluding remarks and closing remarks are represented. The drawbacks of this research are illustrated. Some benefits and some advantages of this research are highlighted. \\ This research uses some approaches to make Extreme SuperHyperGraphs more understandable. In this endeavor, two SuperHyperNotions are defined on the SuperHyperSpace. For that sake in the second definition, the main definition of the Extreme SuperHyperGraph is redefined on the position of the alphabets. Based on the new definition for the Extreme SuperHyperGraph, the new SuperHyperNotion, Extreme SuperHyperSpace, finds the convenient background to implement some results based on that. Some SuperHyperClasses and some Extreme SuperHyperClasses are the cases of this research on the modeling of the regions where are under the attacks of the cancer to recognize this disease as it's mentioned on the title ``Cancer's Recognitions''. To formalize the instances on the SuperHyperNotion, SuperHyperSpace, the new SuperHyperClasses and SuperHyperClasses, are introduced. Some general results are gathered in the section on the SuperHyperSpace and the Extreme SuperHyperSpace. The clarifications, instances and literature reviews have taken the whole way through. In this research, the literature reviews have fulfilled the lines containing the notions and the results. The SuperHyperGraph and Extreme SuperHyperGraph are the SuperHyperModels on the ``Cancer's Recognitions'' and both bases are the background of this research. Sometimes the cancer has been happened on the region, full of cells, groups of cells and embedded styles. In this segment, the SuperHyperModel proposes some SuperHyperNotions based on the connectivities of the moves of the cancer in the longest and strongest styles with the formation of the design and the architecture are formally called `` SuperHyperSpace'' in the themes of jargons and buzzwords. The prefix ``SuperHyper'' refers to the theme of the embedded styles to figure out the background for the SuperHyperNotions. \begin{table}[ht] \centering \caption{An Overlook On This Research And Beyond} \label{136TBLTBL} \begin{tabular}[t]{|c|c|} \hline \textcolor{black}{Advantages}&\textcolor{black}{Limitations}\\ \hline \textcolor{black}{1. }\textcolor{red}{Redefining Extreme SuperHyperGraph} &\textcolor{black}{1. }\textcolor{blue}{General Results} \\ & \\ \textcolor{black}{2. }\textcolor{red}{ SuperHyperSpace}& \\ & \\ \textcolor{black}{3. } \textcolor{red}{Extreme SuperHyperSpace} &\textcolor{black}{2. } \textcolor{blue}{Other SuperHyperNumbers} \\& \\ \textcolor{black}{4. }\textcolor{red}{Modeling of Cancer's Recognitions} & \\& \\ \textcolor{black}{5. }\textcolor{red}{SuperHyperClasses} &\textcolor{black}{3. }\textcolor{blue}{SuperHyperFamilies} \\ \hline \end{tabular} \end{table} In the Table \eqref{136TBLTBL}, benefits and avenues for this research are, figured out, pointed out and spoken out. \section{ Extreme SuperHyperDuality But As The Extensions Excerpt From Dense And Super Forms} \begin{definition}(Different Extreme Types of Extreme SuperHyperDuality).\\ Assume an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider 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{Extreme e-SuperHyperDuality} if $\forall E_i\in E',~\exists E_j\in E_{ESHG:(V,E)}\setminus E'$ such that $V_a\in E_i,E_j;$ \item[$(ii)$] \textbf{Extreme re-SuperHyperDuality} if $\forall E_i\in E',~\exists E_j\in E_{ESHG:(V,E)}\setminus E'$ such that $V_a\in E_i,E_j$ and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$ \item[$(iii)$] \textbf{Extreme v-SuperHyperDuality} if $\forall V_i\in V',~\exists V_j\in V_{ESHG:(V,E)}\setminus V'$ such that $V_i,V_j\in E_a;$ \item[$(iv)$] \textbf{Extreme rv-SuperHyperDuality} if $\forall V_i\in V',~\exists V_j\in V_{ESHG:(V,E)}\setminus V'$ such that $V_i,V_j\in E_a$ and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$ \item[$(v)$] \textbf{Extreme SuperHyperDuality} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality. \end{itemize} \end{definition} \begin{definition}((Extreme) SuperHyperDuality).\\ Assume an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider 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 Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality; \item[$(ii)$] a \textbf{Extreme SuperHyperDuality} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for a Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality; \item[$(iii)$] an \textbf{Extreme SuperHyperDuality SuperHyperPolynomial} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality; and the Extreme power is corresponded to its Extreme coefficient; \item[$(iv)$] a \textbf{Extreme SuperHyperDuality SuperHyperPolynomial} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality; and the Extreme power is corresponded to its Extreme coefficient; \item[$(v)$] an \textbf{Extreme R-SuperHyperDuality} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the consecutive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality; \item[$(vi)$] a \textbf{Extreme R-SuperHyperDuality} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality; \item[$(vii)$] an \textbf{Extreme R-SuperHyperDuality SuperHyperPolynomial} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality; and the Extreme power is corresponded to its Extreme coefficient; \item[$(viii)$] a \textbf{Extreme SuperHyperDuality SuperHyperPolynomial} if it's either of Extreme e-SuperHyperDuality, Extreme re-SuperHyperDuality, Extreme v-SuperHyperDuality, and Extreme rv-SuperHyperDuality and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality consecutive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperDuality; and the Extreme power is corresponded to its Extreme coefficient. \end{itemize} \end{definition} \begin{example}\label{136EXM1} Assume an Extreme SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E)$ in the mentioned Extreme Figures in every Extreme items. \begin{itemize} \item On the Figure \eqref{136NSHG1}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward. $E_1$ and $E_3$ are some empty Extreme SuperHyperEdges but $E_2$ is a loop Extreme SuperHyperEdge and $E_4$ is an Extreme SuperHyperEdge. Thus in the terms of Extreme SuperHyperNeighbor, there's only one Extreme SuperHyperEdge, namely, $E_4.$ The Extreme SuperHyperVertex, $V_3$ is Extreme isolated means that there's no Extreme SuperHyperEdge has it as an Extreme endpoint. Thus the Extreme SuperHyperVertex, $V_3,$ \underline{\textbf{is}} excluded in every given Extreme SuperHyperDuality. \begin{eqnarray*} && \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality}}=\{E_4\}. \\&& \mathcal{C}(NSHG)_{\text{Extreme SuperHyperDuality SuperHyperPolynomial}}=z. \\&& \mathcal{C}(NSHG)_{\text{Extreme R-SuperHyperDuality}}=\{V_4\}. \\&& \mathcal{C}(NSHG)_{{\small\text{Extreme R-SuperHyperDuality SuperHyperPolynomial}}}=3z. \end{eqnarray*} \item On the Figure \eqref{136NSHG2}, the Extreme SuperHyperNotion, namely, Extreme SuperHyperDuality, is up. The Extreme Algorithm is Extremely straightforward. $E_1,E_2$ and $E_3$ are some empty Extreme SuperHyperEdges but $E_4$ is an Extreme SuperHyperEdge. Thus in the terms of Extreme SuperHyperNeighbor, there's only one Extreme SuperHyperEdge, na