A Vizing-type result for semi-total domination

2023

Preprint

In this research, new setting is introduced for new SuperHyperNotions, namely, an 1-failed SuperHyperForcing and Neutrosophic 1-failed SuperHyperForcing. Assume a SuperHyperGraph. Then an ``1-failed SuperHyperForcing'' $\mathcal{Z}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum cardinality of a SuperHyperSet $S$ of black SuperHyperVertices (whereas SuperHyperVertices in $V(G) \setminus S$ are colored white) such that $V(G)$ isn't turned black after finitely many applications of ``the color-change rule'': a white SuperHyperVertex is converted to a black SuperHyperVertex if it is the only white SuperHyperNeighbor of a black SuperHyperVertex. The additional condition is referred by ``1-'' about the usage of any black SuperHyperVertex only once to act on white SuperHyperVertex to be black SuperHyperVertex; a ``neutrosophic 1-failed SuperHyperForcing'' $\mathcal{Z}_n(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum neutrosophic cardinality of a SuperHyperSet $S$ of black SuperHyperVertices (whereas SuperHyperVertices in $V(G) \setminus S$ are colored white) such that $V(G)$ isn't turned black after finitely many applications of ``the color-change rule'': a white SuperHyperVertex is converted to a black SuperHyperVertex if it is the only white SuperHyperNeighbor of a black SuperHyperVertex. The additional condition is referred by ``1-'' about the usage of any black SuperHyperVertex only once to act on white SuperHyperVertex to be black SuperHyperVertex. Assume a SuperHyperGraph. Then an ``$\delta-$1-failed SuperHyperForcing'' is a \underline{maximal} 1-failed SuperHyperForcing of SuperHyperVertices with \underline{maximum} cardinality such that either of the following expressions hold for the (neutrosophic) cardinalities of SuperHyperNeighbors of $s\in S:$ $~|S\cap N(s)| > |S\cap (V\setminus N(s))|+\delta,~|S\cap N(s)| < |S\cap (V\setminus N(s))|+\delta.$ The first Expression, holds if $S$ is an ``$\delta-$SuperHyperOffensive''. And the second Expression, holds if $S$ is an ``$\delta-$SuperHyperDefensive''; a``neutrosophic $\delta-$1-failed SuperHyperForcing'' is a \underline{maximal} neutrosophic 1-failed SuperHyperForcing of SuperHyperVertices with \underline{maximum} neutrosophic cardinality such that either of the following expressions hold for the neutrosophic cardinalities of SuperHyperNeighbors of $s\in S:$ $~|S\cap N(s)|_{neutrosophic} > |S\cap (V\setminus N(s))|_{neutrosophic}+\delta,~ |S\cap N(s)|_{neutrosophic} < |S\cap (V\setminus N(s))|_{neutrosophic}+\delta.$ The first Expression, holds if $S$ is a ``neutrosophic $\delta-$SuperHyperOffensive''. And the second Expression, holds if $S$ is a ``neutrosophic $\delta-$SuperHyperDefensive''. A basic familiarity with SuperHyperGraph theory and neutrosophic SuperHyperGraph theory are proposed.

“…[19] by G. Argiroffo et al (2022), a Vizing-type result for semi-total domination in Ref. [21] by J. Asplund et al (2020), total domination cover rubbling in Ref.…”

Section: Introductionmentioning

confidence: 98%

“…of the SuperHyperVertices of the connected SuperHyperPath N SHP : (V, E), in the SuperHyperModel (21), is the 1-failed SuperHyperForcing.…”

mentioning

confidence: 99%

(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs

2023

Preprint

“…[40] by G. Argiroffo et al (2022), a Vizing-type result for semi-total domination in Ref. [42] by J. Asplund et al (2020), total domination cover rubbling in Ref.…”

mentioning

confidence: 98%

Uncertainty On The Act And Effect Of Cancer Alongside The Foggy Positions Of Cells Toward Neutrosophic Failed SuperHyperClique inside Neutrosophic SuperHyperGraphs Titled Cancer's Recognition

2023

Preprint

In this research, assume a SuperHyperGraph. Then a ``Failed SuperHyperClique'' $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum cardinality of a SuperHyperSet $S$ of SuperHyperVertices such that there's a SuperHyperVertex to have a SuperHyperEdge in common. Assume a SuperHyperGraph. Then an ``$\delta-$Failed SuperHyperClique'' is a \underline{maximal} Failed SuperHyperClique of SuperHyperVertices with \underline{maximum} cardinality such that either of the following expressions hold for the (neutrosophic) cardinalities of SuperHyperNeighbors of $s\in S:$ $~|S\cap N(s)| > |S\cap (V\setminus N(s))|+\delta,~|S\cap N(s)| < |S\cap (V\setminus N(s))|+\delta.$ The first Expression, holds if $S$ is an ``$\delta-$SuperHyperOffensive''. And the second Expression, holds if $S$ is an ``$\delta-$SuperHyperDefensive''; a``neutrosophic $\delta-$Failed SuperHyperClique'' is a \underline{maximal} neutrosophic Failed SuperHyperClique of SuperHyperVertices with \underline{maximum} neutrosophic cardinality such that either of the following expressions hold for the neutrosophic cardinalities of SuperHyperNeighbors of $s\in S:$ $~|S\cap N(s)|_{neutrosophic} > |S\cap (V\setminus N(s))|_{neutrosophic}+\delta,~ |S\cap N(s)|_{neutrosophic} < |S\cap (V\setminus N(s))|_{neutrosophic}+\delta.$ The first Expression, holds if $S$ is a ``neutrosophic $\delta-$SuperHyperOffensive''. And the second Expression, holds if $S$ is a ``neutrosophic $\delta-$SuperHyperDefensive''. A basic familiarity with Neutrosophic Failed SuperHyperClique theory, SuperHyperGraphs theory, and Neutrosophic SuperHyperGraphs theory are proposed.

“…[41] by G. Argiroffo et al (2022), a Vizing-type result for semi-total domination in Ref. [43] by J. Asplund et al (2020), total domination cover rubbling in Ref.…”

mentioning

confidence: 98%

The Shift Paradigm To Classify Separately The Cells and Affected Cells Toward The Totality Under Cancer's Recognition By New Multiple Definitions On the Sets Polynomials Alongside Numbers In The (Neutrosophic) SuperHyperMatching Theory Based on SuperHyperGraph and Neutrosophic SuperHyperGraph

2023

Preprint

In this research, assume a SuperHyperGraph. Then an extreme SuperHyperMatching $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum cardinality of a SuperHyperSet $S$ of high cardinality SuperHyperEdges such that there's no SuperHyperVertex not to in a SuperHyperEdge and there's no SuperHyperEdge to have a SuperHyperVertex in a SuperHyperEdge; a neutrosophic SuperHyperMatching $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum neutrosophic cardinality of a neutrosophic SuperHyperSet $S$ of high neutrosophic cardinality neutrosophic SuperHyperEdges such that there's no neutrosophic SuperHyperVertex not to in a neutrosophic SuperHyperEdge and there's no neutrosophic SuperHyperEdge to have a neutrosophic SuperHyperVertex in a neutrosophic SuperHyperEdge; an extreme SuperHyperMatching SuperHyperPolynomial $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the extreme SuperHyperPolynomial contains the coefficients defined as the number of the maximum cardinality of a SuperHyperSet $S$ of high cardinality SuperHyperEdges such that there's no SuperHyperVertex not to in a SuperHyperEdge and there's no SuperHyperEdge to have a SuperHyperVertex in a SuperHyperEdge and the power is corresponded to its coefficient; a neutrosophic SuperHyperMatching SuperHyperPolynomial $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the neutrosophic SuperHyperPolynomial contains the neutrosophic coefficients defined as the neutrosophic number of the maximum neutrosophic cardinality of a neutrosophic SuperHyperSet $S$ of high neutrosophic cardinality neutrosophic SuperHyperEdges such that there's no neutrosophic SuperHyperVertex not to in a neutrosophic SuperHyperEdge and there's no neutrosophic SuperHyperEdge to have a neutrosophic SuperHyperVertex in a neutrosophic SuperHyperEdge and the neutrosophic power is neutrosophicly corresponded to its neutrosophic coefficient; an extreme R-SuperHyperMatching $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum cardinality of a SuperHyperSet $S$ of high cardinality SuperHyperVertices such that there's no SuperHyperVertex not to in a SuperHyperEdge and there's no SuperHyperEdge to have a SuperHyperVertex in a SuperHyperEdge; a neutrosophic R-SuperHyperMatching $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum neutrosophic cardinality of a neutrosophic SuperHyperSet $S$ of high neutrosophic cardinality neutrosophic SuperHyperVertices such that there's no neutrosophic SuperHyperVertex not to in a neutrosophic SuperHyperEdge and there's no neutrosophic SuperHyperEdge to have a neutrosophic SuperHyperVertex in a neutrosophic SuperHyperEdge; an extreme R-SuperHyperMatching SuperHyperPolynomial $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the extreme SuperHyperPolynomial contains the coefficients defined as the number of the maximum cardinality of a SuperHyperSet $S$ of high cardinality SuperHyperVertices such that there's no SuperHyperVertex not to in a SuperHyperEdge and there's no SuperHyperEdge to have a SuperHyperVertex in a SuperHyperEdge and the power is corresponded to its coefficient; a neutrosophic R-SuperHyperMatching SuperHyperPolynomial $\mathcal{C}(NSHG)$ for a neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the neutrosophic SuperHyperPolynomial contains the neutrosophic coefficients defined as the neutrosophic number of the maximum neutrosophic cardinality of a neutrosophic SuperHyperSet $S$ of high neutrosophic cardinality neutrosophic SuperHyperVertices such that there's no neutrosophic SuperHyperVertex not to in a neutrosophic SuperHyperEdge and there's no neutrosophic SuperHyperEdge to have a neutrosophic SuperHyperVertex in a neutrosophic SuperHyperEdge and the neutrosophic power is neutrosophicly corresponded to its neutrosophic coefficient. Assume a SuperHyperGraph. Then $\delta-$SuperHyperMatching is a maximal of SuperHyperVertices with a maximum cardinality such that either of the following expressions hold for the (neutrosophic) cardinalities of SuperHyperNeighbors of $s\in S:$ there are $|S\cap N(s)| > |S\cap (V\setminus N(s))|+\delta;$ and $ |S\cap N(s)| < |S\cap (V\setminus N(s))|+\delta.$ The first Expression, holds if $S$ is an $\delta-$SuperHyperOffensive. And the second Expression, holds if $S$ is an $\delta-$SuperHyperDefensive; a neutrosophic $\delta-$SuperHyperMatching is a maximal neutrosophic of SuperHyperVertices with maximum neutrosophic cardinality such that either of the following expressions hold for the neutrosophic cardinalities of SuperHyperNeighbors of $s\in S$ there are: $|S\cap N(s)|_{neutrosophic} > |S\cap (V\setminus N(s))|_{neutrosophic}+\delta;$ and $ |S\cap N(s)|_{neutrosophic} < |S\cap (V\setminus N(s))|_{neutrosophic}+\delta.$ The first Expression, holds if $S$ is a neutrosophic $\delta-$SuperHyperOffensive. And the second Expression, holds if $S$ is a neutrosophic $\delta-$SuperHyperDefensive It's useful to define a ``neutrosophic'' version of a SuperHyperMatching . Since there's more ways to get type-results to make a SuperHyperMatching more understandable. For the sake of having neutrosophic SuperHyperMatching, there's a need to ``redefine'' the notion of a ``SuperHyperMatching ''. A basic familiarity with Extreme SuperHyperMatching theory, SuperHyperGraphs, and Neutrosophic SuperHyperGraphs theory are proposed.