SuperHyperForcing
Authors/Creators
Description
Abstract
In this research book, there are two research chapters “Extreme SuperHyper- 1 Forcing” and “Neutrosophic SuperHyperForcing” about some researches on 2 SuperHyperForcing and neutrosophic SuperHyperForcing. With researches 3 on the basic properties, the research book starts to make SuperHyperForcing 4 theory and neutrosophic SuperHyperForcing theory more understandable. 5
6 In the first chapter, in this research, new setting is introduced for new 7 SuperHyperNotions, namely, SuperHyperForcing and Neutrosophic SuperHy- 8 perForcing. Two different types of SuperHyperDefinitions are debut for them 9 but the research goes further and the SuperHyperNotion, SuperHyperUniform, 10 and SuperHyperClass based on that are well-defined and well-reviewed. The 11 literature review is implemented in the whole of this research. For shining the 12 elegancy and the significancy of this research, the comparison between this 13 SuperHyperNotion with other SuperHyperNotions and fundamental SuperHy- 14 perNumbers are featured. The definitions are followed by the examples and the 15 instances thus the clarifications are driven with different tools. The applications 16 are figured out to make sense about the theoretical aspect of this ongoing 17 research. The “Cancer’s Recognitions” are the under research to figure out the 18 challenges make sense about ongoing and upcoming research. The special case 19 is up. The cells are viewed in the deemed ways. There are different types of 20 them. Some of them are individuals and some of them are well-modeled by the 21 group of cells. These types are all officially called “SuperHyperVertex” but the 22 relations amid them all officially called “SuperHyperEdge”. The frameworks 23 “SuperHyperGraph” and “neutrosophic SuperHyperGraph” are chosen and elec- 24 ted to research about “Cancer’s Recognitions”. Thus these complex and dense 25 SuperHyperModels open up some avenues to research on theoretical segments 26 and “Cancer’s Recognitions”. Some avenues are posed to pursue this research. 27 It’s also officially collected in the form of some questions and some problems. 28 Assume a SuperHyperGraph. Then a “SuperHyperForcing” Z(NSHG) for a 29 neutrosophic SuperHyperGraph NSHG : (V,E) is the minimum cardinality of 30 a SuperHyperSet S of black SuperHyperVertices (whereas SuperHyperVertices 31 in V (G) \ S are colored white) such that V (G) is turned black after finitely 32 many applications of “the color-change rule”: a white SuperHyperVertex is 33 converted to a black SuperHyperVertex if it is the only white SuperHyper- 34 Neighbor of a black SuperHyperVertex; a “neutrosophic SuperHyperForcing” 35 Zn(NSHG) for a neutrosophic SuperHyperGraph NSHG : (V,E) is the 36 minimum neutrosophic cardinality of a SuperHyperSet S of black SuperHy- 37
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Abstract
perVertices (whereas SuperHyperVertices in V (G) \ S are colored white) such 38 that V (G) is turned black after finitely many applications of “the color-change 39 rule”: a white SuperHyperVertex is converted to a black SuperHyperVertex 40 if it is the only white SuperHyperNeighbor of a black SuperHyperVertex. 41 Assume a SuperHyperGraph. Then a “SuperHyperForcing” Z(NSHG) for a 42 neutrosophic SuperHyperGraph NSHG : (V,E) is the minimum cardinality of 43 a SuperHyperSet S of black SuperHyperVertices (whereas SuperHyperVertices 44 in V (G)\S are colored white) such that V (G) is turned black after finitely many 45 applications of “the color-change rule”: a white SuperHyperVertex is converted 46 to a black SuperHyperVertex if it is the only white SuperHyperNeighbor of 47 a black SuperHyperVertex; a “neutrosophic SuperHyperForcing” Zn(NSHG) 48 for a neutrosophic SuperHyperGraph NSHG : (V,E) is the minimum neutro- 49 sophic cardinality of a SuperHyperSet S of black SuperHyperVertices (whereas 50 SuperHyperVertices in V (G) \ S are colored white) such that V (G) is turned 51 black after finitely many applications of “the color-change rule”: a white Super- 52 HyperVertex is converted to a black SuperHyperVertex if it is the only white 53 SuperHyperNeighbor of a black SuperHyperVertex. Assume a SuperHyper- 54 Graph. Then an “δ−SuperHyperForcing” is a minimal SuperHyperForcing of 55 SuperHyperVertices with minimum cardinality such that either of the following 56 expressions hold for the (neutrosophic) cardinalities of SuperHyperNeighbors of 57 s∈S: |S∩N(s)|>|S∩(V \N(s))|+δ, |S∩N(s)|<|S∩(V \N(s))|+δ. 58 The first Expression, holds if S is an “δ−SuperHyperOffensive”. And the 59 second Expression, holds if S is an “δ−SuperHyperDefensive”; a“neutrosophic 60 δ−SuperHyperForcing” is a minimal neutrosophic SuperHyperForcing of Su- 61 perHyperVertices with minimum neutrosophic cardinality such that either of 62 the following expressions hold for the neutrosophic cardinalities of SuperHyper- 63 Neighbors of s ∈ S : |S∩N(s)|neutrosophic > |S∩(V \N(s))|neutrosophic+δ, |S∩ 64 N (s)|neutrosophic < |S ∩ (V \ N (s))|neutrosophic + δ. The first Expression, holds 65 if S is a “neutrosophic δ−SuperHyperOffensive”. And the second Expression, 66 holds if S is a “neutrosophic δ−SuperHyperDefensive”. It’s useful to define 67 “neutrosophic” version of SuperHyperForcing. Since there’s more ways to get 68 type-results to make SuperHyperForcing more understandable. For the sake of 69 having neutrosophic SuperHyperForcing, there’s a need to “redefine” the notion 70 of “SuperHyperForcing”. The SuperHyperVertices and the SuperHyperEdges 71 are assigned by the labels from the letters of the alphabets. In this procedure, 72 there’s the usage of the position of labels to assign to the values. Assume 73 a SuperHyperForcing. It’s redefined neutrosophic SuperHyperForcing if the 74 mentioned Table holds, concerning, “The Values of Vertices, SuperVertices, 75 Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic Super- 76 HyperGraph” with the key points, “The Values of The Vertices & The Number 77 of Position in Alphabet”, “The Values of The SuperVertices&The Minimum 78 Values of Its Vertices”, “The Values of The Edges&The Minimum Values of 79 Its Vertices”, “The Values of The HyperEdges&The Minimum Values of Its 80 Vertices”, “The Values of The SuperHyperEdges&The Minimum Values of Its 81Endpoints”. To get structural examples and instances, I’m going to introduce 82 the next SuperHyperClass of SuperHyperGraph based on SuperHyperForcing. 83 It’s the main. It’ll be disciplinary to have the foundation of previous definition in 84 the kind of SuperHyperClass. If there’s a need to have all SuperHyperConnectiv- 85 ities until the SuperHyperForcing, then it’s officially called “SuperHyperForcing” 86 but otherwise, it isn’t SuperHyperForcing. There are some instances about the 87
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clarifications for the main definition titled “SuperHyperForcing”. These two 88 examples get more scrutiny and discernment since there are characterized in 89 the disciplinary ways of the SuperHyperClass based on SuperHyperForcing. 90 For the sake of having neutrosophic SuperHyperForcing, there’s a need to 91 “redefine” the notion of “neutrosophic SuperHyperForcing” and “neutrosophic 92 SuperHyperForcing”. The SuperHyperVertices and the SuperHyperEdges are 93 assigned by the labels from the letters of the alphabets. In this procedure, 94 there’s the usage of the position of labels to assign to the values. Assume 95 a neutrosophic SuperHyperGraph. It’s redefined “neutrosophic SuperHyper- 96 Graph” if the intended Table holds. And SuperHyperForcing are redefined 97 “neutrosophic SuperHyperForcing” if the intended Table holds. It’s useful to 98 define “neutrosophic” version of SuperHyperClasses. Since there’s more ways 99 to get neutrosophic type-results to make neutrosophic SuperHyperForcing 100 more understandable. Assume a neutrosophic SuperHyperGraph. There are 101 some neutrosophic SuperHyperClasses if the intended Table holds. Thus 102 SuperHyperPath, SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, 103 SuperHyperMultiPartite, and SuperHyperWheel, are “neutrosophic SuperHy- 104 perPath”, “neutrosophic SuperHyperCycle”, “neutrosophic SuperHyperStar”, 105 “neutrosophic SuperHyperBipartite”, “neutrosophic SuperHyperMultiPartite”, 106 and “neutrosophic SuperHyperWheel” if the intended Table holds. A SuperHy- 107 perGraph has “neutrosophic SuperHyperForcing” where it’s the strongest [the 108 maximum neutrosophic value from all SuperHyperForcing amid the maximum 109value amid all SuperHyperVertices from a SuperHyperForcing.] SuperHyperFor- 110 cing. A graph is SuperHyperUniform if it’s SuperHyperGraph and the number 111 of elements of SuperHyperEdges are the same. Assume a neutrosophic SuperHy- 112 perGraph. There are some SuperHyperClasses as follows. It’s SuperHyperPath 113 if it’s only one SuperVertex as intersection amid two given SuperHyperEdges 114 with two exceptions; it’s SuperHyperCycle if it’s only one SuperVertex as 115 intersection amid two given SuperHyperEdges; it’s SuperHyperStar it’s only one 116 SuperVertex as intersection amid all SuperHyperEdges; it’s SuperHyperBipart- 117 ite it’s only one SuperVertex as intersection amid two given SuperHyperEdges 118and these SuperVertices, forming two separate sets, has no SuperHyperEdge in 119 common; it’s SuperHyperMultiPartite it’s only one SuperVertex as intersection 120 amid two given SuperHyperEdges and these SuperVertices, forming multi 121 separate sets, has no SuperHyperEdge in common; it’s SuperHyperWheel if it’s 122 only one SuperVertex as intersection amid two given SuperHyperEdges and one 123 SuperVertex has one SuperHyperEdge with any common SuperVertex. The 124 SuperHyperModel proposes the specific designs and the specific architectures. 125 The SuperHyperModel is officially called “SuperHyperGraph” and “Neutro- 126 sophic SuperHyperGraph”. In this SuperHyperModel, The “specific” cells and 127 “specific group” of cells are SuperHyperModeled as “SuperHyperVertices” and 128 the common and intended properties between “specific” cells and “specific 129 group” of cells are SuperHyperModeled as “SuperHyperEdges”. Sometimes, 130 it’s useful to have some degrees of determinacy, indeterminacy, and neutrality 131 to have more precise SuperHyperModel which in this case the SuperHyper- 132 Model is called “neutrosophic”. In the future research, the foundation will 133 be based on the “Cancer’s Recognitions” and the results and the definitions 134 will be introduced in redeemed ways. The recognition of the cancer in the 135 long-term function. The specific region has been assigned by the model [it’s 136 called SuperHyperGraph] and the long cycle of the move from the cancer is 137
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Abstract
identified by this research. Sometimes the move of the cancer hasn’t be easily 138 identified since there are some determinacy, indeterminacy and neutrality about 139 the moves and the effects of the cancer on that region; this event leads us 140 to choose another model [it’s said to be neutrosophic SuperHyperGraph] to 141 have convenient perception on what’s happened and what’s done. There are 142 some specific models, which are well-known and they’ve got the names, and 143 some SuperHyperGeneral SuperHyperModels. The moves and the traces of 144 the cancer on the complex tracks and between complicated groups of cells 145 could be fantasized by a neutrosophic SuperHyperPath(-/SuperHyperCycle, 146 SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHy- 147 perWheel). The aim is to find either the longest SuperHyperForcing or the 148 strongest SuperHyperForcing in those neutrosophic SuperHyperModels. For 149 the longest SuperHyperForcing, called SuperHyperForcing, and the strongest 150SuperHyperCycle, called neutrosophic SuperHyperForcing, some general results 151 are introduced. Beyond that in SuperHyperStar, all possible SuperHyperPaths 152 have only two SuperHyperEdges but it’s not enough since it’s essential to have 153 at least three SuperHyperEdges to form any style of a SuperHyperCycle. There 154 isn’t any formation of any SuperHyperCycle but literarily, it’s the deformation 155 of any SuperHyperCycle. It, literarily, deforms and it doesn’t form. A basic 156 familiarity with SuperHyperGraph theory and neutrosophic SuperHyperGraph 157 theory are proposed. 158
Keywords: (Neutrosophic) SuperHyperGraph, (Neutrosophic) SuperHyper- 159
Forcing, Cancer’s Recognitions 160 AMS Subject Classification: 05C17, 05C22, 05E45 161 162 In the second chapter, in this research, new setting is introduced for new 163 SuperHyperNotions, namely, SuperHyperForcing and Neutrosophic SuperHy- 164 perForcing. Two different types of SuperHyperDefinitions are debut for them 165 but the research goes further and the SuperHyperNotion, SuperHyperUniform, 166 and SuperHyperClass based on that are well-defined and well-reviewed. The 167 literature review is implemented in the whole of this research. For shining the 168 elegancy and the significancy of this research, the comparison between this 169 SuperHyperNotion with other SuperHyperNotions and fundamental SuperHy- 170perNumbers are featured. The definitions are followed by the examples and the 171 instances thus the clarifications are driven with different tools. The applications 172 are figured out to make sense about the theoretical aspect of this ongoing 173 research. The “Cancer’s Recognitions” are the under research to figure out the 174 challenges make sense about ongoing and upcoming research. The special case 175 is up. The cells are viewed in the deemed ways. There are different types of 176 them. Some of them are individuals and some of them are well-modeled by the 177 group of cells. These types are all officially called “SuperHyperVertex” but the 178 relations amid them all officially called “SuperHyperEdge”. The frameworks 179“SuperHyperGraph” and “neutrosophic SuperHyperGraph” are chosen and elec- 180 ted to research about “Cancer’s Recognitions”. Thus these complex and dense 181 SuperHyperModels open up some avenues to research on theoretical segments 182 and “Cancer’s Recognitions”. Some avenues are posed to pursue this research. 183 It’s also officially collected in the form of some questions and some problems. 184 Assume a SuperHyperGraph. Then a “SuperHyperForcing” Z(NSHG) for a 185 neutrosophic SuperHyperGraph NSHG : (V,E) is the minimum cardinality of 186
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a SuperHyperSet S of black SuperHyperVertices (whereas SuperHyperVertices 187 in V (G) \ S are colored white) such that V (G) is turned black after finitely 188 many applications of “the color-change rule”: a white SuperHyperVertex is 189 converted to a black SuperHyperVertex if it is the only white SuperHyper- 190 Neighbor of a black SuperHyperVertex; a “neutrosophic SuperHyperForcing” 191 Zn(NSHG) for a neutrosophic SuperHyperGraph NSHG : (V,E) is the 192 minimum neutrosophic cardinality of a SuperHyperSet S of black SuperHy- 193 perVertices (whereas SuperHyperVertices in V (G) \ S are colored white) such 194 that V (G) is turned black after finitely many applications of “the color-change 195 rule”: a white SuperHyperVertex is converted to a black SuperHyperVertex 196 if it is the only white SuperHyperNeighbor of a black SuperHyperVertex. 197 Assume a SuperHyperGraph. Then a “SuperHyperForcing” Z(NSHG) for a 198 neutrosophic SuperHyperGraph NSHG : (V,E) is the minimum cardinality of 199 a SuperHyperSet S of black SuperHyperVertices (whereas SuperHyperVertices 200 in V (G)\S are colored white) such that V (G) is turned black after finitely many 201 applications of “the color-change rule”: a white SuperHyperVertex is converted 202 to a black SuperHyperVertex if it is the only white SuperHyperNeighbor of 203 a black SuperHyperVertex; a “neutrosophic SuperHyperForcing” Zn(NSHG) 204 for a neutrosophic SuperHyperGraph NSHG : (V, E) is the minimum neutro- 205 sophic cardinality of a SuperHyperSet S of black SuperHyperVertices (whereas 206 SuperHyperVertices in V (G) \ S are colored white) such that V (G) is turned 207 black after finitely many applications of “the color-change rule”: a white Super- 208 HyperVertex is converted to a black SuperHyperVertex if it is the only white 209 SuperHyperNeighbor of a black SuperHyperVertex. Assume a SuperHyper- 210Graph. Then an “δ−SuperHyperForcing” is a minimal SuperHyperForcing of 211 SuperHyperVertices with minimum cardinality such that either of the following 212 expressions hold for the (neutrosophic) cardinalities of SuperHyperNeighbors of 213 s∈S: |S∩N(s)|>|S∩(V \N(s))|+δ, |S∩N(s)|<|S∩(V \N(s))|+δ. 214 The first Expression, holds if S is an “δ−SuperHyperOffensive”. And the 215 second Expression, holds if S is an “δ−SuperHyperDefensive”; a“neutrosophic 216 δ−SuperHyperForcing” is a minimal neutrosophic SuperHyperForcing of Su- 217 perHyperVertices with minimum neutrosophic cardinality such that either of 218 the following expressions hold for the neutrosophic cardinalities of SuperHyper- 219Neighbors of s ∈ S : |S∩N(s)|neutrosophic > |S∩(V \N(s))|neutrosophic+δ, |S∩ 220 N (s)|neutrosophic < |S ∩ (V \ N (s))|neutrosophic + δ. The first Expression, holds 221 if S is a “neutrosophic δ−SuperHyperOffensive”. And the second Expression, 222 holds if S is a “neutrosophic δ−SuperHyperDefensive”. It’s useful to define 223 “neutrosophic” version of SuperHyperForcing. Since there’s more ways to get 224 type-results to make SuperHyperForcing more understandable. For the sake of 225 having neutrosophic SuperHyperForcing, there’s a need to “redefine” the notion 226 of “SuperHyperForcing”. The SuperHyperVertices and the SuperHyperEdges 227 are assigned by the labels from the letters of the alphabets. In this procedure, 228 there’s the usage of the position of labels to assign to the values. Assume 229 a SuperHyperForcing. It’s redefined neutrosophic SuperHyperForcing if the 230 mentioned Table holds, concerning, “The Values of Vertices, SuperVertices, 231 Edges, HyperEdges, and SuperHyperEdges Belong to The Neutrosophic Super- 232HyperGraph” with the key points, “The Values of The Vertices & The Number 233 of Position in Alphabet”, “The Values of The SuperVertices&The Minimum 234 Values of Its Vertices”, “The Values of The Edges&The Minimum Values of 235 Its Vertices”, “The Values of The HyperEdges&The Minimum Values of Its 236
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Vertices”, “The Values of The SuperHyperEdges&The Minimum Values of Its 237 Endpoints”. To get structural examples and instances, I’m going to introduce 238 the next SuperHyperClass of SuperHyperGraph based on SuperHyperForcing. 239 It’s the main. It’ll be disciplinary to have the foundation of previous definition in 240 the kind of SuperHyperClass. If there’s a need to have all SuperHyperConnectiv- 241 ities until the SuperHyperForcing, then it’s officially called “SuperHyperForcing” 242 but otherwise, it isn’t SuperHyperForcing. There are some instances about the 243 clarifications for the main definition titled “SuperHyperForcing”. These two 244 examples get more scrutiny and discernment since there are characterized in 245 the disciplinary ways of the SuperHyperClass based on SuperHyperForcing. 246 For the sake of having neutrosophic SuperHyperForcing, there’s a need to 247 “redefine” the notion of “neutrosophic SuperHyperForcing” and “neutrosophic 248 SuperHyperForcing”. The SuperHyperVertices and the SuperHyperEdges are 249 assigned by the labels from the letters of the alphabets. In this procedure, 250 there’s the usage of the position of labels to assign to the values. Assume 251 a neutrosophic SuperHyperGraph. It’s redefined “neutrosophic SuperHyper- 252 Graph” if the intended Table holds. And SuperHyperForcing are redefined 253 “neutrosophic SuperHyperForcing” if the intended Table holds. It’s useful to 254 define “neutrosophic” version of SuperHyperClasses. Since there’s more ways 255 to get neutrosophic type-results to make neutrosophic SuperHyperForcing 256 more understandable. Assume a neutrosophic SuperHyperGraph. There are 257 some neutrosophic SuperHyperClasses if the intended Table holds. Thus 258SuperHyperPath, SuperHyperCycle, SuperHyperStar, SuperHyperBipartite, 259 SuperHyperMultiPartite, and SuperHyperWheel, are “neutrosophic SuperHy- 260 perPath”, “neutrosophic SuperHyperCycle”, “neutrosophic SuperHyperStar”, 261 “neutrosophic SuperHyperBipartite”, “neutrosophic SuperHyperMultiPartite”, 262 and “neutrosophic SuperHyperWheel” if the intended Table holds. A SuperHy- 263 perGraph has “neutrosophic SuperHyperForcing” where it’s the strongest [the 264 maximum neutrosophic value from all SuperHyperForcing amid the maximum 265 value amid all SuperHyperVertices from a SuperHyperForcing.] SuperHyperFor- 266 cing. A graph is SuperHyperUniform if it’s SuperHyperGraph and the number 267of elements of SuperHyperEdges are the same. Assume a neutrosophic SuperHy- 268 perGraph. There are some SuperHyperClasses as follows. It’s SuperHyperPath 269 if it’s only one SuperVertex as intersection amid two given SuperHyperEdges 270 with two exceptions; it’s SuperHyperCycle if it’s only one SuperVertex as 271 intersection amid two given SuperHyperEdges; it’s SuperHyperStar it’s only one 272 SuperVertex as intersection amid all SuperHyperEdges; it’s SuperHyperBipart- 273 ite it’s only one SuperVertex as intersection amid two given SuperHyperEdges 274 and these SuperVertices, forming two separate sets, has no SuperHyperEdge in 275 common; it’s SuperHyperMultiPartite it’s only one SuperVertex as intersection 276 amid two given SuperHyperEdges and these SuperVertices, forming multi 277 separate sets, has no SuperHyperEdge in common; it’s SuperHyperWheel if it’s 278 only one SuperVertex as intersection amid two given SuperHyperEdges and one 279 SuperVertex has one SuperHyperEdge with any common SuperVertex. The 280 SuperHyperModel proposes the specific designs and the specific architectures. 281 The SuperHyperModel is officially called “SuperHyperGraph” and “Neutro- 282 sophic SuperHyperGraph”. In this SuperHyperModel, The “specific” cells and 283 “specific group” of cells are SuperHyperModeled as “SuperHyperVertices” and 284 the common and intended properties between “specific” cells and “specific 285 group” of cells are SuperHyperModeled as “SuperHyperEdges”. Sometimes, 286
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it’s useful to have some degrees of determinacy, indeterminacy, and neutrality 287 to have more precise SuperHyperModel which in this case the SuperHyper- 288 Model is called “neutrosophic”. In the future research, the foundation will 289 be based on the “Cancer’s Recognitions” and the results and the definitions 290 will be introduced in redeemed ways. The recognition of the cancer in the 291 long-term function. The specific region has been assigned by the model [it’s 292 called SuperHyperGraph] and the long cycle of the move from the cancer is 293 identified by this research. Sometimes the move of the cancer hasn’t be easily 294 identified since there are some determinacy, indeterminacy and neutrality about 295 the moves and the effects of the cancer on that region; this event leads us 296 to choose another model [it’s said to be neutrosophic SuperHyperGraph] to 297 have convenient perception on what’s happened and what’s done. There are 298 some specific models, which are well-known and they’ve got the names, and 299 some SuperHyperGeneral SuperHyperModels. The moves and the traces of 300 the cancer on the complex tracks and between complicated groups of cells 301could be fantasized by a neutrosophic SuperHyperPath(-/SuperHyperCycle, 302 SuperHyperStar, SuperHyperBipartite, SuperHyperMultipartite, SuperHy- 303 perWheel). The aim is to find either the longest SuperHyperForcing or the 304 strongest SuperHyperForcing in those neutrosophic SuperHyperModels. For 305 the longest SuperHyperForcing, called SuperHyperForcing, and the strongest 306 SuperHyperCycle, called neutrosophic SuperHyperForcing, some general results 307 are introduced. Beyond that in SuperHyperStar, all possible SuperHyperPaths 308 have only two SuperHyperEdges but it’s not enough since it’s essential to have 309 at least three SuperHyperEdges to form any style of a SuperHyperCycle. There 310isn’t any formation of any SuperHyperCycle but literarily, it’s the deformation 311 of any SuperHyperCycle. It, literarily, deforms and it doesn’t form. A basic 312 familiarity with SuperHyperGraph theory and neutrosophic SuperHyperGraph 313 theory are proposed. 314
Keywords: (Neutrosophic) SuperHyperGraph, (Neutrosophic) SuperHyper- 315
Forcing, Cancer’s Recognitions 316 AMS Subject Classification: 05C17, 05C22, 05E45 317 318 The following references are cited by chapters. 319 320[Ref1] Henry Garrett, “Basic Notions on (Neutrosophic) SuperHyper- 321 Forcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recogni- 322tions And (Neutrosophic) SuperHyperGraphs”, ResearchGate 2022, (doi: 323
10.13140/RG.2.2.11369.16487).
The links to the contributions of this research book are listed below.
324 325 326 327
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[Ref1] Henry Garrett, “Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs”, ResearchGate 2022, (doi: 10.13140/RG.2.2.11369.16487).
328 Article #99 329 330 Basic Notions on (Neutrosophic) SuperHyperForcing And (Neutrosophic) 331 SuperHyperModeling in Cancer’s Recognitions And (Neutrosophic) SuperHy- 332 perGraphs 333 334 @Wordpress: https://drhenrygarrett.wordpress.com/2022/12/29/superhyperforcin3g3-5 19/ 336 337@Preprints_org: ?????? 338 339
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@ResearchGate: https://www.researchgate.net/publication/366673729 340 341 @Scribd: https://www.scribd.com/document/617025195 342 343@Academia: https://www.academia.edu/93968337 344 345 @Zenodo: https://zenodo.org/record/7493849 346 347 348
[Ref2] Henry Garrett, “(Neutrosophic) 1-Failed SuperHyperForcing in Cancer’s Recognitions And (Neutrosophic) SuperHyperGraphs”, ResearchGate 2022, (doi: ??????).
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Some studies and researches about neutrosophic graphs, are proposed as 350 book in the following by Henry Garrett (2022) which is indexed by Google 351 Scholar and has more than 2455 readers in Scribd. It’s titled “Beyond Neutro- 352 sophic Graphs” and published by Ohio: E-publishing: Educational Publisher 353 1091 West 1st Ave Grandview Heights, Ohio 43212 United State. This research 354 book covers different types of notions and settings in neutrosophic graph theory 355 and neutrosophic SuperHyperGraph theory. 356
[Ref] Henry Garrett, (2022). “Beyond Neutrosophic Graphs”, Ohio: E-publishing: Educational Publisher 1091 West 1st Ave Grandview Heights, Ohio 43212 United States. ISBN: 978-1-59973-725-6 (http://fs.unm.edu/BeyondNeutrosophicGraphs.pdf).
357 358
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[Ref] Henry Garrett, (2022). “Beyond Neutrosophic Graphs”, 360 Ohio: E-publishing: Educational Publisher 1091 West 1st Ave Grand- 361 view Heights, Ohio 43212 United States. ISBN: 978-1-59973-725-6 362
(http://fs.unm.edu/BeyondNeutrosophicGraphs.pdf). 363 364 Also, some studies and researches about neutrosophic graphs, are proposed 365 as book in the following by Henry Garrett (2022) which is indexed by Google 366 Scholar and has more than 3152 readers in Scribd. It’s titled “Neutrosophic 367Duality” and published by Florida: GLOBAL KNOWLEDGE - Publishing 368 House 848 Brickell Ave Ste 950 Miami, Florida 33131 United States. This 369 research book presents different types of notions SuperHyperResolving and 370 SuperHyperDominating in the setting of duality in neutrosophic graph theory 371 and neutrosophic SuperHyperGraph theory. This research book has scrutiny 372 on the complement of the intended set and the intended set, simultaneously. 373 It’s smart to consider a set but acting on its complement that what’s done in 374 this research book which is popular in the terms of high readers in Scribd. 375
[Ref] Henry Garrett, (2022). “Neutrosophic Duality”, Florida: GLOBAL KNOW- LEDGE - Publishing House 848 Brickell Ave Ste 950 Miami, Florida 33131 United States. ISBN: 978-1-59973-743-0 (http://fs.unm.edu/NeutrosophicDuality.pdf).
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[Ref] Henry Garrett, (2022). “Neutrosophic Duality”, Florida: 377 GLOBAL KNOWLEDGE - Publishing House 848 Brickell Ave Ste 378 950 Miami, Florida 33131 United States. ISBN: 978-1-59973-743-0 379
(http://fs.unm.edu/NeutrosophicDuality.pdf).
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Abstract
Background 381
There are some researches covering the topic of this research. In what follows, 382 there are some discussion and literature reviews about them. 383 First article is titled “properties of SuperHyperGraph and neutrosophic Super- 384 HyperGraph” in Ref. [HG1] by Henry Garrett (2022). It’s first step toward the 385 research on neutrosophic SuperHyperGraphs. This research article is published 386 on the journal “Neutrosophic Sets and Systems” in issue 49 and the pages 531- 387 561. In this research article, different types of notions like dominating, resolving, 388 coloring, Eulerian(Hamiltonian) neutrosophic path, n-Eulerian(Hamiltonian) 389 neutrosophic path, zero forcing number, zero forcing neutrosophic- number, 390 independent number, independent neutrosophic-number, clique number, clique 391 neutrosophic-number, matching number, matching neutrosophic-number, girth, 392neutrosophic girth, 1-zero-forcing number, 1-zero- forcing neutrosophic-number, 393 failed 1-zero-forcing number, failed 1-zero-forcing neutrosophic-number, global- 394 offensive alliance, t-offensive alliance, t-defensive alliance, t-powerful alliance, 395 and global-powerful alliance are defined in SuperHyperGraph and neutrosophic 396 SuperHyperGraph. Some Classes of SuperHyperGraph and Neutrosophic Su- 397 perHyperGraph are cases of research. Some results are applied in family of 398 SuperHyperGraph and neutrosophic SuperHyperGraph. Thus this research 399 article has concentrated on the vast notions and introducing the majority of 400 notions. 401 The seminal paper and groundbreaking article is titled “neutrosophic co-degree 402 and neutrosophic degree alongside chromatic numbers in the setting of some 403 classes related to neutrosophic hypergraphs” in Ref. [HG2] by Henry Garrett 404 (2022). In this research article, a novel approach is implemented on SuperHy- 405 perGraph and neutrosophic SuperHyperGraph based on general forms without 406 using neutrosophic classes of neutrosophic SuperHyperGraph. It’s published in 407 prestigious and fancy journal is entitled “Journal of Current Trends in Com- 408 puter Science Research (JCTCSR)” with abbreviation “J Curr Trends Comp 409 Sci Res” in volume 1 and issue 1 with pages 06-14. The research article studies 410 deeply with choosing neutrosophic hypergraphs instead of neutrosophic Super- 411 HyperGraph. It’s the breakthrough toward independent results based on initial 412 background. 413 In some articles are titled “(Neutrosophic) SuperHyperModeling of Cancer’s Re- 414 cognitions Featuring (Neutrosophic) SuperHyperDefensive SuperHyperAlliances” 415 in Ref. [HG3] by Henry Garrett (2022), “(Neutrosophic) SuperHyperAlliances 416 With SuperHyperDefensive and SuperHyperOffensive Type-SuperHyperSet On 417(Neutrosophic) SuperHyperGraph With (Neutrosophic) SuperHyperModeling 418 of Cancer’s Recognitions And Related (Neutrosophic) SuperHyperClasses” in 419 Ref. [HG4] by Henry Garrett (2022), “SuperHyperGirth on SuperHyperGraph 420 and Neutrosophic SuperHyperGraph With SuperHyperModeling of Cancer’s 421 Recognitions” in Ref. [HG5] by Henry Garrett (2022), “Some SuperHy- 422 perDegrees and Co-SuperHyperDegrees on Neutrosophic SuperHyperGraphs 423 and SuperHyperGraphs Alongside Applications in Cancer’s Treatments” in 424 Ref. [HG6] by Henry Garrett (2022), “SuperHyperDominating and Super- 425 HyperResolving on Neutrosophic SuperHyperGraphs And Their Directions in 426 Game Theory and Neutrosophic SuperHyperClasses” in Ref. [HG7] by Henry 427 Garrett (2022), “Basic Notions on (Neutrosophic) SuperHyperForcing And 428 (Neutrosophic) SuperHyperModeling in Cancer’s Recognitions And (Neutro- 429
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sophic) SuperHyperGraphs” in Ref. [HG8] by Henry Garrett (2022), “Basic 430 Neutrosophic Notions Concerning SuperHyperDominating and Neutrosophic 431 SuperHyperResolving in SuperHyperGraph” in Ref. [HG9] by Henry Gar- 432 rett (2022), “Initial Material of Neutrosophic Preliminaries to Study Some 433 Neutrosophic Notions Based on Neutrosophic SuperHyperEdge (NSHE) in 434 Neutrosophic SuperHyperGraph (NSHG)” in Ref. [HG10] by Henry Garrett 435 (2022), there are some endeavors to formalize the basic SuperHyperNotions 436 about neutrosophic SuperHyperGraph and SuperHyperGraph. 437 Some studies and researches about neutrosophic graphs, are proposed as book 438 in Ref. [HG11] by Henry Garrett (2022) which is indexed by Google Scholar 439 and has more than 2455 readers in Scribd. It’s titled “Beyond Neutrosophic 440 Graphs” and published by Ohio: E-publishing: Educational Publisher 1091 441 West 1st Ave Grandview Heights, Ohio 43212 United State. This research book 442 covers different types of notions and settings in neutrosophic graph theory and 443 neutrosophic SuperHyperGraph theory. 444 Also, some studies and researches about neutrosophic graphs, are proposed as 445 book in Ref. [HG12] by Henry Garrett (2022) which is indexed by Google 446 Scholar and has more than 3152 readers in Scribd. It’s titled “Neutrosophic 447 Duality” and published by Florida: GLOBAL KNOWLEDGE - Publishing 448 House 848 Brickell Ave Ste 950 Miami, Florida 33131 United States. This 449 research book presents different types of notions SuperHyperResolving and 450SuperHyperDominating in the setting of duality in neutrosophic graph theory 451 and neutrosophic SuperHyperGraph theory. This research book has scrutiny 452 on the complement of the intended set and the intended set, simultaneously. 453 It’s smart to consider a set but acting on its complement that what’s done in 454
this research book which is popular in the terms of high readers in Scribd.
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[2]
[3]
[4]
[5]
[6]
[7]
Bibliography 456
Henry Garrett, “Properties of SuperHyperGraph and Neut- 457 rosophic SuperHyperGraph”, Neutrosophic Sets and Sys- 458 tems 49 (2022) 531-561 (doi: 10.5281/zenodo.6456413). 459 (http://fs.unm.edu/NSS/NeutrosophicSuperHyperGraph34.pdf). 460(https://digitalrepository.unm.edu/nss_journal/vol49/iss1/34). 461
Henry Garrett, “Neutrosophic Co-degree and Neutrosophic Degree along- 462 side Chromatic Numbers in the Setting of Some Classes Related to Neut- 463 rosophic Hypergraphs”, J Curr Trends Comp Sci Res 1(1) (2022) 06-14. 464
Henry Garrett, “”, Preprints 2022, 2022120549 (doi: 10.20944/pre- 465 prints202212.0549.v1). 466
Henry Garrett, “(Neutrosophic) SuperHyperModeling of Cancer’s 467 Recognitions Featuring (Neutrosophic) SuperHyperDefensive Super- 468HyperAlliances”, Preprints 2022, 2022120549 (doi: 10.20944/pre- 469 prints202212.0549.v1). 470
Henry Garrett, “(Neutrosophic) SuperHyperAl liances With SuperHyperDe- 471 fensive and SuperHyperOffensive Type-SuperHyperSet On (Neutrosophic) 472 SuperHyperGraph With (Neutrosophic) SuperHyperModeling of Cancer’s 473 Recognitions And Related (Neutrosophic) SuperHyperClasses”, Preprints 474 2022, 2022120540 (doi: 10.20944/preprints202212.0540.v1). 475
Henry Garrett, “SuperHyperGirth on SuperHyperGraph and Neutrosophic 476 SuperHyperGraph With SuperHyperModeling of Cancer’s Recognitions”, 477 Preprints 2022, 2022120500 (doi: 10.20944/preprints202212.0500.v1). 478
Henry Garrett, “Some SuperHyperDegrees and Co-SuperHyperDegrees 479 on Neutrosophic SuperHyperGraphs and SuperHyperGraphs Alongside 480Applications in Cancer’s Treatments”, Preprints 2022, 2022120324 (doi: 481 10.20944/preprints202212.0324.v1). 482
Henry Garrett, “SuperHyperDominating and SuperHyperResolving on 483 Neutrosophic SuperHyperGraphs And Their Directions in Game Theory 484and Neutrosophic SuperHyperClasses”, Preprints 2022, 2022110576 (doi: 485
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10.20944/preprints202211.0576.v1).
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Bibliography
[8] Henry Garrett, “Basic Notions on (Neutrosophic) SuperHyperFor- 487 cing And (Neutrosophic) SuperHyperModeling in Cancer’s Recogni- 488 tions And (Neutrosophic) SuperHyperGraphs”, ResearchGate 2022, (doi: 489 10.13140/RG.2.2.11369.16487). 490
HG8
HG9
[9]
[10]
[11]
[12]
[13]
[14] [15] [16] [17]
Henry Garrett, “Basic Neutrosophic Notions Concerning SuperHyper- 491 Dominating and Neutrosophic SuperHyperResolving in SuperHyperGraph”, 492 ResearchGate 2022 (doi: 10.13140/RG.2.2.29173.86244). 493
Henry Garrett, “Initial Material of Neutrosophic Preliminaries to Study 494 Some Neutrosophic Notions Based on Neutrosophic SuperHyperEdge 495(NSHE) in Neutrosophic SuperHyperGraph (NSHG)”, ResearchGate 2022 496 (doi: 10.13140/RG.2.2.25385.88160). 497
Henry Garrett, (2022). “Beyond Neutrosophic Graphs”, Ohio: E- 498 publishing: Educational Publisher 1091 West 1st Ave Grand- 499 view Heights, Ohio 43212 United States. ISBN: 979-1-59973-725-6 500 (http://fs.unm.edu/BeyondNeutrosophicGraphs.pdf). 501
Henry Garrett, (2022). “Neutrosophic Duality”, Florida: GLOBAL 502 KNOWLEDGE - Publishing House 848 Brickell Ave Ste 950 503 Miami, Florida 33131 United States. ISBN: 978-1-59973-743-0 504 (http://fs.unm.edu/NeutrosophicDuality.pdf). 505
F. Smarandache, “Extension of HyperGraph to n-SuperHyperGraph and to 506 Plithogenic n-SuperHyperGraph, and Extension of HyperAlgebra to n-ary 507 (Classical-/Neutro-/Anti-) HyperAlgebra”, Neutrosophic Sets and Systems 508 33 (2020) 290-296. (doi: 10.5281/zenodo.3783103). 509
M. Akram et al., “Single-valued neutrosophic Hypergraphs”, TWMS J. 510 App. Eng. Math. 8 (1) (2018) 122-135. 511
S. Broumi et al., “Single-valued neutrosophic graphs”, Journal of New 512 Theory 10 (2016) 86-101. 513
H. Wang et al., “Single-valued neutrosophic sets”, Multispace and 514 Multistructure 4 (2010) 410-413. 515
H.T. Nguyen and E.A. Walker, “A First course in fuzzy logic”, CRC 516 Press, 2006. 517
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| Book #63
| Title: SuperHyperForcing
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Bibliography
In this research book, there are two research chapters “Extreme SuperHyperForcing” and “Neutrosophic SuperHyperForcing” about some researches on SuperHyperForcing and
neutrosophic SuperHyperForcing.
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