Neutrosophic 1-Failed SuperHyperForcing in the SuperHyperFunction To Use Neutrosophic SuperHyperGraphs on Cancer’s Neutrosophic Recognition And Beyond
Authors: Garrett Henry 
Published: January 2023

In this research, new setting is introduced for new SuperHyperNotion, namely, 11 Neutrosophic 1-failed SuperHyperForcing. Two different types of SuperHyperDefinitions 12 are debut for them but the research goes further and the SuperHyperNotion, 13 SuperHyperUniform, and SuperHyperClass based on that are well-defined and 14 well-reviewed. The literature review is implemented in the whole of this research. For 15 shining the elegancy and the significancy of this research, the comparison between this 16 SuperHyperNotion with other SuperHyperNotions and fundamental 17 SuperHyperNumbers are featured. The definitions are followed by the examples and the 18 instances thus the clarifications are driven with different tools. The applications are 19 figured out to make sense about the theoretical aspect of this ongoing research. The 20 &ldquo;Cancer&rsquo;s Neutrosophic Recognition&rdquo; are the under research to figure out the challenges 21 make sense about ongoing and upcoming research. The special case is up. The cells are 22 viewed in the deemed ways. There are different types of them. Some of them are 23 individuals and some of them are well-modeled by the group of cells. These types are all 24 officially called &ldquo;SuperHyperVertex&rdquo; but the relations amid them all officially called 25 &ldquo;SuperHyperEdge&rdquo;. The frameworks &ldquo;SuperHyperGraph&rdquo; and &ldquo;neutrosophic 26 SuperHyperGraph&rdquo; are chosen and elected to research about &ldquo;Cancer&rsquo;s Neutrosophic 27 Recognition&rdquo;. Thus these complex and dense SuperHyperModels open up some avenues 28 to research on theoretical segments and &ldquo;Cancer&rsquo;s Neutrosophic Recognition&rdquo;. Some 29 avenues are posed to pursue this research. It&rsquo;s also officially collected in the form of 30 some questions and some problems. Assume a SuperHyperGraph. Then a &ldquo;1-failed 31 SuperHyperForcing&rdquo; Z(NSHG) for a neutrosophic SuperHyperGraph NSHG : (V,E) 32 is the maximum cardinality of a SuperHyperSet S of black SuperHyperVertices 33 (whereas SuperHyperVertices in V (G) \ S are colored white) such that V (G) isn&rsquo;t 34 turned black after finitely many applications of &ldquo;the color-change rule&rdquo;: a white 35 SuperHyperVertex is converted to a black SuperHyperVertex if it is the only white 36 SuperHyperNeighbor of a black SuperHyperVertex. The additional condition is referred 37 by &ldquo;1-&rdquo; about the usage of any black SuperHyperVertex only once to act on white 38 SuperHyperVertex to be black SuperHyperVertex; a &ldquo;neutrosophic 1-failed 39 SuperHyperForcing&rdquo; Zn(NSHG) for a neutrosophic SuperHyperGraph NSHG : (V,E) 40 is the maximum neutrosophic cardinality of a SuperHyperSet S of black 41 SuperHyperVertices (whereas SuperHyperVertices in V (G) \ S are colored white) such 42 that V (G) isn&rsquo;t turned black after finitely many applications of &ldquo;the color-change rule&rdquo;: 43 a white SuperHyperVertex is converted to a black SuperHyperVertex if it is the only 44 1/128 white SuperHyperNeighbor of a black SuperHyperVertex. The additional condition is 45 referred by &ldquo;1-&rdquo; about the usage of any black SuperHyperVertex only once to act on 46 white SuperHyperVertex to be black SuperHyperVertex. Assume a SuperHyperGraph. 47 Then an &ldquo;&delta;&minus;1-failed SuperHyperForcing&rdquo; is a maximal 1-failed SuperHyperForcing of 48 SuperHyperVertices with maximum cardinality such that either of the following 49 expressions hold for the (neutrosophic) cardinalities of SuperHyperNeighbors of s &isin; S : 50 |S &cap;N(s)| &gt; |S &cap;(V \N(s))|+&delta;, |S &cap;N(s)| &lt; |S &cap;(V \N(s))|+&delta;. The first Expression, 51 holds if S is an &ldquo;&delta;&minus;SuperHyperOffensive&rdquo;. And the second Expression, holds if S is an 52 &ldquo;&delta;&minus;SuperHyperDefensive&rdquo;; a&ldquo;neutrosophic &delta;&minus;1-failed SuperHyperForcing&rdquo; is a maximal 53 neutrosophic 1-failed SuperHyperForcing of SuperHyperVertices with maximum 54 neutrosophic cardinality such that either of the following expressions hold for the 55 neutrosophic cardinalities of SuperHyperNeighbors of s &isin; S : |S &cap; N(s)|neutrosophic &gt; 56 |S &cap; (V \ N (s))|neutrosophic + &delta;, |S &cap; N (s)|neutrosophic &lt; |S &cap; (V \ N (s))|neutrosophic + &delta;. 57 The first Expression, holds if S is a &ldquo;neutrosophic &delta;&minus;SuperHyperOffensive&rdquo;. And the 58 second Expression, holds if S is a &ldquo;neutrosophic &delta;&minus;SuperHyperDefensive&rdquo;. It&rsquo;s useful to 59 define &ldquo;neutrosophic&rdquo; version of 1-failed SuperHyperForcing. Since there&rsquo;s more ways to 60 get type-results to make 1-failed SuperHyperForcing more understandable. For the sake 61 of having neutrosophic 1-failed SuperHyperForcing, there&rsquo;s a need to &ldquo;redefine&rdquo; the 62 notion of &ldquo;1-failed SuperHyperForcing&rdquo;. The SuperHyperVertices and the 63 SuperHyperEdges are assigned by the labels from the letters of the alphabets. In this 64 procedure, there&rsquo;s the usage of the position of labels to assign to the values. Assume a 65 1-failed SuperHyperForcing. It&rsquo;s redefined neutrosophic 1-failed SuperHyperForcing if 66 the mentioned Table holds, concerning, &ldquo;The Values of Vertices, SuperVertices, Edges, 67 HyperEdges, and SuperHyperEdges Belong to The Neutrosophic SuperHyperGraph&rdquo; 68 with the key points, &ldquo;The Values of The Vertices &amp; The Number of Position in 69 Alphabet&rdquo;, &ldquo;The Values of The SuperVertices&amp;The maximum Values of Its Vertices&rdquo;, 70 &ldquo;The Values of The Edges&amp;The maximum Values of Its Vertices&rdquo;, &ldquo;The Values of The 71 HyperEdges&amp;The maximum Values of Its Vertices&rdquo;, &ldquo;The Values of The 72 SuperHyperEdges&amp;The maximum Values of Its Endpoints&rdquo;. To get structural examples 73 and instances, I&rsquo;m going to introduce the next SuperHyperClass of SuperHyperGraph 74 based on 1-failed SuperHyperForcing. It&rsquo;s the main. It&rsquo;ll be disciplinary to have the 75 foundation of previous definition in the kind of SuperHyperClass. If there&rsquo;s a need to 76 have all SuperHyperConnectivities until the 1-failed SuperHyperForcing, then it&rsquo;s 77 officially called &ldquo;1-failed SuperHyperForcing&rdquo; but otherwise, it isn&rsquo;t 1-failed 78 SuperHyperForcing. There are some instances about the clarifications for the main 79 definition titled &ldquo;1-failed SuperHyperForcing&rdquo;. These two examples get more scrutiny 80 and discernment since there are characterized in the disciplinary ways of the 81 SuperHyperClass based on 1-failed SuperHyperForcing. For the sake of having 82 neutrosophic 1-failed SuperHyperForcing, there&rsquo;s a need to &ldquo;redefine&rdquo; the notion of 83 &ldquo;neutrosophic 1-failed SuperHyperForcing&rdquo; and &ldquo;neutrosophic 1-failed 84 SuperHyperForcing&rdquo;. The SuperHyperVertices and the SuperHyperEdges are assigned 85 by the labels from the letters of the alphabets. In this procedure, there&rsquo;s the usage of 86 the position of labels to assign to the values. Assume a neutrosophic SuperHyperGraph. 87 It&rsquo;s redefined &ldquo;neutrosophic SuperHyperGraph&rdquo; if the intended Table holds. And 88 1-failed SuperHyperForcing are redefined &ldquo;neutrosophic 1-failed SuperHyperForcing&rdquo; if 89 the intended Table holds. It&rsquo;s useful to define &ldquo;neutrosophic&rdquo; version of 90 SuperHyperClasses. Since there&rsquo;s more ways to get neutrosophic type-results to make 91 neutrosophic 1-failed SuperHyperForcing more understandable. Assume a neutrosophic 92 SuperHyperGraph. There are some neutrosophic SuperHyperClasses if the intended 93 Table holds. Thus SuperHyperPath, SuperHyperCycle, SuperHyperStar, 94 SuperHyperBipartite, SuperHyperMultiPartite, and SuperHyperWheel, are 95 &ldquo;neutrosophic SuperHyperPath&rdquo;, &ldquo;neutrosophic SuperHyperCycle&rdquo;, &ldquo;neutrosophic 96 SuperHyperStar&rdquo;, &ldquo;neutrosophic SuperHyperBipartite&rdquo;, &ldquo;neutrosophic 97 2/128 SuperHyperMultiPartite&rdquo;, and &ldquo;neutrosophic SuperHyperWheel&rdquo; if the intended Table 98 holds. A SuperHyperGraph has &ldquo;neutrosophic 1-failed SuperHyperForcing&rdquo; where it&rsquo;s 99 the strongest [the maximum neutrosophic value from all 1-failed SuperHyperForcing 100 amid the maximum value amid all SuperHyperVertices from a 1-failed 101 SuperHyperForcing.] 1-failed SuperHyperForcing. A graph is SuperHyperUniform if it&rsquo;s 102 SuperHyperGraph and the number of elements of SuperHyperEdges are the same. 103 Assume a neutrosophic SuperHyperGraph. There are some SuperHyperClasses as 104 follows. It&rsquo;s SuperHyperPath if it&rsquo;s only one SuperVertex as intersection amid two given 105 SuperHyperEdges with two exceptions; it&rsquo;s SuperHyperCycle if it&rsquo;s only one 106 SuperVertex as intersection amid two given SuperHyperEdges; it&rsquo;s SuperHyperStar it&rsquo;s 107 only one SuperVertex as intersection amid all SuperHyperEdges; it&rsquo;s 108 SuperHyperBipartite it&rsquo;s only one SuperVertex as intersection amid two given 109 SuperHyperEdges and these SuperVertices, forming two separate sets, has no 110 SuperHyperEdge in common; it&rsquo;s SuperHyperMultiPartite it&rsquo;s only one SuperVertex as 111 intersection amid two given SuperHyperEdges and these SuperVertices, forming multi 112 separate sets, has no SuperHyperEdge in common; it&rsquo;s SuperHyperWheel if it&rsquo;s only one 113 SuperVertex as intersection amid two given SuperHyperEdges and one SuperVertex has 114 one SuperHyperEdge with any common SuperVertex. The SuperHyperModel proposes 115 the specific designs and the specific architectures. The SuperHyperModel is officially 116 called &ldquo;SuperHyperGraph&rdquo; and &ldquo;Neutrosophic SuperHyperGraph&rdquo;. In this 117 SuperHyperModel, The &ldquo;specific&rdquo; cells and &ldquo;specific group&rdquo; of cells are 118 SuperHyperModeled as &ldquo;SuperHyperVertices&rdquo; and the common and intended properties 119 between &ldquo;specific&rdquo; cells and &ldquo;specific group&rdquo; of cells are SuperHyperModeled as 120 &ldquo;SuperHyperEdges&rdquo;. Sometimes, it&rsquo;s useful to have some degrees of determinacy, 121 indeterminacy, and neutrality to have more precise SuperHyperModel which in this case 122 the SuperHyperModel is called &ldquo;neutrosophic&rdquo;. In the future research, the foundation 123 will be based on the &ldquo;Cancer&rsquo;s Neutrosophic Recognition&rdquo; and the results and the 124 definitions will be introduced in redeemed ways. The neutrosophic recognition of the 125 cancer in the long-term function. The specific region has been assigned by the model 126 [it&rsquo;s called SuperHyperGraph] and the long cycle of the move from the cancer is 127 identified by this research. Sometimes the move of the cancer hasn&rsquo;t be easily identified 128 since there are some determinacy, indeterminacy and neutrality about the moves and 129 the effects of the cancer on that region; this event leads us to choose another model [it&rsquo;s 130 said to be neutrosophic SuperHyperGraph] to have convenient perception on what&rsquo;s 131 happened and what&rsquo;s done. There are some specific models, which are well-known and 132 they&rsquo;ve got the names, and some SuperHyperGeneral SuperHyperModels. The moves 133 and the traces of the cancer on the complex tracks and between complicated groups of 134 cells could be fantasized by a neutrosophic SuperHyperPath(-/SuperHyperCycle, 135 SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHyperWheel). 136 The aim is to find either the longest 1-failed SuperHyperForcing or the strongest 137 1-failed SuperHyperForcing in those neutrosophic SuperHyperModels. For the longest 138 1-failed SuperHyperForcing, called 1-failed SuperHyperForcing, and the strongest 139 SuperHyperCycle, called neutrosophic 1-failed SuperHyperForcing, some general results 140 are introduced. Beyond that in SuperHyperStar, all possible SuperHyperPaths have 141 only two SuperHyperEdges but it&rsquo;s not enough since it&rsquo;s essential to have at least three 142 SuperHyperEdges to form any style of a SuperHyperCycle. There isn&rsquo;t any formation of 143 any SuperHyperCycle but literarily, it&rsquo;s the deformation of any SuperHyperCycle. It, 144 literarily, deforms and it doesn&rsquo;t form. A basic familiarity with SuperHyperGraph 145 theory and neutrosophic SuperHyperGraph theory are proposed.