Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction to Use Neutrosophic SuperHyperGraphs on Cancers Neutrosophic Recognition and Beyond

Abstract

Henry Garrett

In this research, new setting is introduced for new SuperHyperNotion, namely, Neutrosophic 1-failed SuperHyperForc- ing. Two different types of SuperHyperDefinitions are debut for them but the research goes further and the SuperHyper- Notion, 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 “Can- cer’s Neutrosophic 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 “Su- perHyperGraph” and “neutrosophic SuperHyperGraph” are chosen and elected to research about “Cancer’s Neutro- sophic Recognition”. Thus these complex and dense SuperHyperModels open up some avenues to research on theoreti- cal segments and “Cancer’s Neutrosophic Recognition”. Some avenues are posed to pursue this research. It’s also officially collected in the form of some questions and some problems. Assume a SuperHyperGraph. Then a “1-failed SuperHyperForcing” Z(NSHG) for a neutrosophic SuperHyperGraph is the maximum cardinality of a SuperHyperSet S of black SuperHyperVertices (whereas SuperHyperVertices in V (G) \ 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” Zn (NSHG) for a neutrosophic SuperHyperGraph is the maximum neutrosophic cardinality of a SuperHyperSet S of black SuperHyperVertices (whereas SuperHyperVertices in V (G) \ 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 SuperHyperVer- tex only once to act on white SuperHyperVertex to be black SuperHyperVertex. Assume a SuperHyperGraph. Then an “δ−1-failed SuperHyperForcing” is a maximal 1-failed SuperHyperForcing of SuperHyperVertices with maximum car- dinality such that either of the following expressions hold for the (neutrosophic) cardinalities of SuperHyperNeighbors of |S ∩N(s)| > |S ∩(V \N(s))|+δ, |S ∩N(s)| < |S ∩(V \N(s))|+δ. The first Expression, holds if S is an “δ−SuperHyperOf- fensive”. And the second Expression, holds if S is an “δ−SuperHyperDefensive”; a“neutrosophic δ−1-failed SuperHy- perForcing” is a maxim neutrosophic 1-failed SuperHyperForcing of SuperHyperVertices with maximum neutrosophic cardinality such that either of the following expressions hold for the neutrosophic cardinalities of SuperHyperNeighbors of s ∈ S : |S ∩ N(s)|neutrosophic > S ∩(V \N(s))|neutrosophic +δ, |S ∩N(s)|neutrosophic < |S ∩(V \N(s))|neutrosophic +δ. The first Expres- sion, holds if S is a “neutrosophic δ−SuperHyperOffensive”. And the second Expression, holds if S is a “neutrosophic δ−SuperHyperDefensive”. It’s useful to define “neutrosophic” version of 1-failed SuperHyperForcing. Since there’s more ways to get type-results to make 1-failed SuperHyperForcing more understandable. For the sake of having neutro- sophic 1-failed SuperHyperForcing, there’s a need to “redefine” the notion of “1-failed SuperHyperForcing”. The Su- perHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this proce- dure, there’s the usage of the position of labels to assign to the values. Assume a 1-failed SuperHyperForcing. It’s redefined neutrosophic 1-failed SuperHyperForcing if the mentioned Table holds, concerning, “The Values of Vertices,  SuperVertices, Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph” with the key points, “The Values of The Vertices & The Number of Position in Alphabet”, “The Values of The SuperVertices&The maximum Values of Its Vertices”, “The Values of The Edges&The maximum Values of Its Vertices”, “The Values of The HyperEdges&The maximum Values of Its Vertices”, “The Values of The SuperHyperEdges&The maximum Values of Its Endpoints”. To get structural examples and instances, I’m going to introduce the next SuperHyperClass of SuperHyper- Graph based on 1-failed SuperHyperForcing. It’s the main. It’ll be disciplinary to have the foundation of previous defi- nition in the kind of SuperHyperClass. If there’s a need to have all SuperHyperConnectivities until the 1-failed Super- HyperForcing, then it’s officially called “1-failed SuperHyperForcing” but otherwise, it isn’t 1-failed SuperHyperForcing. There are some instances about the clarifications for the main definition titled “1-failed SuperHy- perForcing”. These two examples get more scrutiny and discernment since there are characterized in the disciplinary ways of the SuperHyperClass based on 1-failed SuperHyperForcing. For the sake of having neutrosophic 1-failed Supe- rHyperForcing, there’s a need to “redefine” the notion of “neutrosophic 1-failed SuperHyperForcing” and “neutro- sophic 1-failed SuperHyperForcing”. The SuperHyperVertices and the SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this procedure, there’s the usage of the position of labels to assign to the values. Assume a neutrosophic SuperHyperGraph. It’s redefined “neutrosophic SuperHyperGraph” if the intended Table holds. And 1-failed SuperHyperForcing are redefined “neutrosophic 1-failed SuperHyperForcing” if the intended Table holds. It’s useful to define “neutrosophic” version of SuperHyperClasses. Since there’s more ways to get neutrosophic type-re- sults to make neutrosophic 1-failed SuperHyperForcing more understandable. Assume a neutrosophic SuperHyper- Graph. There are some neutrosophic SuperHyperClasses if the intended Table holds. Thus SuperHyperPath, SuperHy- perCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultiPartite, and SuperHyperWheel, are “neutrosophic SuperHyperPath”, “neutrosophic SuperHyperCycle”, “neutrosophic SuperHyperStar”, “neutrosophic SuperHyperBi- partite”, “neutrosophic SuperHyperMultiPartite”, and “neutrosophic SuperHyperWheel” if the intended Table holds. A SuperHyperGraph has “neutrosophic 1-failed SuperHyperForcing” where it’s the strongest [the maximum neutro- sophic value from all 1-failed SuperHyperForcing amid the maximum value amid all SuperHyperVertices from a 1-failed SuperHyperForcing.] 1-failed SuperHyperForcing. A graph is SuperHyperUniform if it’s SuperHyperGraph and the number of elements of SuperHyperEdges are the same. Assume a neutrosophic SuperHyperGraph. There are some Su- perHyperClasses as follows. It’s SuperHyperPath if it’s only one SuperVertex as intersection amid two given SuperHyp- erEdges with two exceptions; it’s SuperHyperCycle if it’s only one SuperVertex as intersection amid two given SuperHy- perEdges; it’s SuperHyperStar it’s only one SuperVertex as intersection amid all SuperHyperEdges; it’s SuperHyperBipartite it’s only one SuperVertex as intersection amid two given SuperHyperEdges and these SuperVerti- ces, forming two separate sets, has no SuperHyperEdge in common; it’s SuperHyperMultiPartite it’s only one SuperVer- tex as intersection amid two given SuperHyperEdges and these SuperVertices, forming multi separate sets, has no Supe- rHyperEdge in common; it’s SuperHyperWheel if it’s only one SuperVertex as intersection amid two given SuperHyperEdges and one SuperVertex has one SuperHyperEdge with any common SuperVertex. The SuperHyperMod- el proposes the specific designs and the specific architectures. The SuperHyperModel is officially called “SuperHyper- Graph” and “Neutrosophic SuperHyperGraph”. In this SuperHyperModel, The “specific” cells and “specific group” of cells are SuperHyperModeled as “SuperHyperVertices” and the common and intended properties between “specific” cells and “specific group” of cells are SuperHyperModeled as “SuperHyperEdges”. Sometimes, it’s useful to have some degrees of determinacy, indeterminacy, and neutrality to have more precise SuperHyperModel which in this case the SuperHyperModel is called “neutrosophic”. In the future research, the foundation will be based on the “Cancer’s Neu- trosophic Recognition” and the results and the definitions will be introduced in redeemed ways. The neutrosophic rec- ognition of the cancer in the long-term function. The specific region has been assigned by the model [it’s called Super- HyperGraph] and the long cycle of the move from the cancer is identified by this research. Sometimes the move of the cancer hasn’t be easily identified since there are some determinacy, indeterminacy and neutrality about the moves and the effects of the cancer on that region; this event leads us to choose another model [it’s said to be neutrosophic Super- HyperGraph] to have convenient perception on what’s happened and what’s done. There are some specific models, which are well-known and they’ve got the names, and some SuperHyperGeneral SuperHyperModels. The moves and the traces of the cancer on the complex tracks and between complicated groups of cells could be fantasized by a neutrosoph- ic SuperHyperPath(-/SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyper- Wheel). The aim is to find either the longest 1-failed SuperHyperForcing or the strongest 1-failed SuperHyperForcing in those neutrosophic SuperHyperModels. For the longest 1-failed SuperHyperForcing, called 1-failed SuperHyper- Forcing, and the strongest SuperHyperCycle, called neutrosophic 1-failed SuperHyperForcing, some general results are introduced. Beyond that in SuperHyperStar, all possible SuperHyperPaths have only two SuperHyperEdges but it’s not enough since it’s essential to have at least three SuperHyperEdges to form any style of a SuperHyperCycle. There isn’t any formation of any SuperHyperCycle but literarily, it’s the deformation of any SuperHyperCycle. It, literarily, deforms and it doesn’t form. A basic familiarity with SuperHyperGraph theory and neutrosophic SuperHyperGraph theory are proposed.

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