# Fuzzy Graphs and Fuzzy Hypergraphs

Leonid S. Bershtein (Taganrog Technological Institute of Southern Federal University, Russia) and Alexander V. Bozhenyuk (Taganrog Technological Institute of Southern Federal University, Russia)
DOI: 10.4018/978-1-59904-849-9.ch105

## Abstract

Graph theory has numerous application to problems in systems analysis, operations research, economics, and transportation. However, in many cases, some aspects of a graph-theoretic problem may be uncertain. For example, the vehicle travel time or vehicle capacity on a road network may not be known exactly. In such cases, it is natural to deal with the uncertainty using the methods of fuzzy sets and fuzzy logic. Hypergraphs (Berge,1989) are the generalization of graphs in case of set of multiarity relations. It means the expansion of graph models for the modeling complex systems. In case of modelling systems with fuzzy binary and multiarity relations between objects, transition to fuzzy hypergraphs, which combine advantages both fuzzy and graph models, is more natural. It allows to realise formal optimisation and logical procedures. However, using of the fuzzy graphs and hypergraphs as the models of various systems (social, economic systems, communication networks and others) leads to difficulties. The graph isomorphic transformations are reduced to redefinition of vertices and edges. This redefinition doesn’t change properties the graph determined by an adjacent and an incidence of its vertices and edges. Fuzzy independent set, domination fuzzy set, fuzzy chromatic set are invariants concerning the isomorphism transformations of the fuzzy graphs and fuzzy hypergraph and allow make theirs structural analysis.
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## Main Definitions Of Fuzzy Graphs And Hypergraphs

This article presents the main notations of fuzzy graphs and fuzzy hypergraphs, invariants of fuzzy graphs and hypergraphs.

## Key Terms in this Chapter

Multiarity Relation: A multiarity relation R between elements of sets A, B, …, C is a subset of A×B×…×C.

Binary Relation: A binary relation R from a set A to a set B is a subset of A×B.

Isomorphic Graphs: Two graphs that have a structure-preserving vertex bijection between them.

Graph Invariant: A property of a graph that is preserved by isomorphisms.

Graph: A graph G = (V, E) is a mathematical structure consisting of two finite sets V and E. The elements of V are called vertices (or nodes), and the elements of E are called edges. Each edge has a set of one or two vertices associated to it, which are called its endpoints.

Binary Symmetric Relation: A relation R on a set A is symmetric if for all x,y?A xRy?yRx.

Fuzzy Set: A generalization of the definition of the classical set. A fuzzy set is characterized by a membership function, which maps the member of the universe into the unit interval, thus assigning to elements of the universe degrees of belongingness with respect to a set.

Membership Function: The membership function of a fuzzy set is a generalization of the characteristic function of crisp sets.

Hypergraph: A hypergraph on a finite set X={x1,x2,…,xn} is a family H={E1,E2,…,Em} of subsets of X such that Ei ? Ø and and .

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Chapter 203
Robots in Education  (pages 1383-1388)
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Chapter 204
Andrzej Rusiecki
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Chapter 205
Kai Keng Ang, Chai Quek
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Chapter 206
Agostino Poggi, Michele Tomaiuolo
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Chapter 207
Chun-Cheng Peng, George D. Magoulas
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Chapter 208
Ioanna Roussaki, Ioannis Papaioannou, Miltiades Anagnostou
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Chapter 209
Fernando Zacarías Flores, Dionicio Zacarías Flores, Rosalba Cuapa Canto, Luis Miguel Guzmán Muñoz
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Chapter 210
Fatih Onur Hocaoglu, Ömer Nezih Gerek, Mehmet Kurban
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Chapter 211
Jesús Bernardino Alonso Hernández, Patricia Henríquez Rodríguez
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Chapter 212
Carlos M. Travieso González, Carlos F. Romero
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Chapter 213
Yan Yan, Yu-Jin Zhang
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Chapter 214
Arturo Berrones, Dexmont Peña, Ricardo Sánchez
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Chapter 215
Mirjam Sepesy Maucec, Zdravko Kacic
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Chapter 216
Snorre Aunet, Hans Kristian Otnes Berge
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Chapter 218
M. Martínez-Zarzuela, F. J. Díaz Pernas, D. González Ortega, J. F. Díez Higuera, M. Antón Rodríguez
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Chapter 219
M. Martínez-Zarzuela, F. J. Díaz Pernas, D. González Ortega, J. F. Díez Higuera, M. Antón Rodríguez
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Chapter 220
Pi-Sheng Deng
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Chapter 221
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Chapter 222
Support Vector Machines  (pages 1518-1523)
Cecilio Angulo, Luis Gonzalez-Abril
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Chapter 223
Ioannis Papaioannou, Ioanna Roussaki, Miltiades Anagnostou
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Chapter 224
Prayag Narula, Sudip Misra, Sanjay Kumar Dhurandher
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Chapter 225
Swarm Robotics  (pages 1537-1542)
Amanda J.C. Sharkey
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Chapter 226
Symbol Grounding Problem  (pages 1543-1548)
Angelo Loula, João Queiroz
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Chapter 227
Symbolic Search  (pages 1549-1554)
Stefan Edelkamp
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Chapter 228
Snorre Aunet
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Chapter 229
Raymond Chiong
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Chapter 230
T. T. Wong, C. W. Leung
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Chapter 231
Fernando Mateo, Ramón J. Aliaga, Jorge D. Martínez, José Ma Monzó, Rafael Gadea
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Chapter 232
Pablo Mesejo, Enrique Fernández-Blanco, Diego Martínez-Feijóo, Francisco J. Blanco
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Chapter 233
Julio J. Valdés, Alan J. Barton
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Chapter 234
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Chapter 235
Alberte Castro Ponte, Gregorio Iglesias, Francisco Taveira Pinto, Rodrigo Carballo
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Chapter 236
Sylvia Encheva, Sharil Tumin
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Chapter 237