Fuzzy Graphs and Fuzzy Hypergraphs

Fuzzy Graphs and Fuzzy Hypergraphs

Michael G. Voskoglou (Graduate T. E. I. of Western Greece, Greece) and Tarasankar Pramanik (Khanpur Gangche High School, India)
DOI: 10.4018/978-1-5225-9380-5.ch019

Abstract

Relationship is the core building block of a network, and today's world advances through the complex networks. Graph theory deals with such problems more efficiently. But whenever vagueness or imprecision arises in such relationships, fuzzy graph theory helps. However, fuzzy hypergraphs are more advanced generalization of fuzzy graphs. Whenever there is a need to define multiary relationship rather than binary relationship, one can use fuzzy hypergraphs. In this chapter, interval-valued fuzzy hypergraph is discussed which is a generalization of fuzzy hypergraph. Several approaches to find shortest path between two given nodes in an interval-valued fuzzy graphs is described here. Many researchers have focused on fuzzy shortest path problem in a network due to its importance to many applications such as communications, routing, transportation, etc.
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Fuzzy Graph

A fuzzy graph 978-1-5225-9380-5.ch019.m03 is a triplet consisting of a non-empty set 978-1-5225-9380-5.ch019.m04 together with a pair of functions 978-1-5225-9380-5.ch019.m05 and 978-1-5225-9380-5.ch019.m06 such that

978-1-5225-9380-5.ch019.m07
for all 978-1-5225-9380-5.ch019.m08

Here the two fuzzy sets 978-1-5225-9380-5.ch019.m09 and 978-1-5225-9380-5.ch019.m10 are called fuzzy vertex set and fuzzy edge set of 978-1-5225-9380-5.ch019.m11 respectively. Clearly, 978-1-5225-9380-5.ch019.m12 is a fuzzy relation on 978-1-5225-9380-5.ch019.m13.

The fuzzy subgraph of a fuzzy graph is a fuzzy graph whose fuzzy set is a subset of the fuzzy set of the given fuzzy graph.

A fuzzy graph 978-1-5225-9380-5.ch019.m14 is said to be a partial fuzzy subgraph of 978-1-5225-9380-5.ch019.m15 if 978-1-5225-9380-5.ch019.m16 and 978-1-5225-9380-5.ch019.m17.

The fuzzy graph 978-1-5225-9380-5.ch019.m18 is called a fuzzy subgraph of 978-1-5225-9380-5.ch019.m19 if 978-1-5225-9380-5.ch019.m20 for all 978-1-5225-9380-5.ch019.m21 and 978-1-5225-9380-5.ch019.m22 for all 978-1-5225-9380-5.ch019.m23 where 978-1-5225-9380-5.ch019.m24.

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