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## Über dieses Buch

This book provides a timely overview of fuzzy graph theory, laying the foundation for future applications in a broad range of areas. It introduces readers to fundamental theories, such as Craine’s work on fuzzy interval graphs, fuzzy analogs of Marczewski’s theorem, and the Gilmore and Hoffman characterization. It also introduces them to the Fulkerson and Gross characterization and Menger’s theorem, the applications of which will be discussed in a forthcoming book by the same authors. This book also discusses in detail important concepts such as connectivity, distance and saturation in fuzzy graphs.

Thanks to the good balance between the basics of fuzzy graph theory and new findings obtained by the authors, the book offers an excellent reference guide for advanced undergraduate and graduate students in mathematics, engineering and computer science, and an inspiring read for all researchers interested in new developments in fuzzy logic and applied mathematics.

## Inhaltsverzeichnis

### Chapter 1. Fuzzy Sets and Relations

Abstract
The notion of a fuzzy graph was initially introduced by Kauffman (Introduction to the theory of fuzzy sets, Academic Press Inc., Orlando) in [91]. However, the development of fuzzy graph theory is due to the ground setting papers of Rosenfeld (Fuzzy sets and their applications, Academic Press, New York) [154] and Yeh and Bang (Fuzzy sets and their applications, Academic Press, New York) [186]. In Rosenfeld’s paper, basic structural and connectivity concepts were presented while Yeh and Bang introduced different connectivity parameters and discussed their application. Rosenfeld obtained the fuzzy analogs of several graph-theoretic concepts like bridges, paths, cycles, trees, and connectedness. Most of the theoretical development of fuzzy graph theory is based on Rosenfeld’s initial work.
Sunil Mathew, John N. Mordeson, Davender S. Malik

### Chapter 2. Fuzzy Graphs

Abstract
A graph represents a particular relationship between elements of a set V. It gives an idea about the extent of the relationship between any two elements of V. We can solve this problem by using a weighted graph if proper weights are known. But in most of the situations, the weights may not be known, and the relationships are ‘fuzzy’ in a natural sense. Hence, a fuzzy relation can deal with the situation in a better way. As an example, if V represents certain locations and a network of roads is to be constructed between elements of V, then the costs of construction of the links are fuzzy. But the costs can be compared, to some extent using the terrain and local factors and can be modeled as fuzzy relations. Thus, fuzzy graph models are more helpful and realistic in natural situations.
Sunil Mathew, John N. Mordeson, Davender S. Malik

### Chapter 3. Connectivity in Fuzzy Graphs

Abstract
In graph theory, edge analysis is not very necessary because all edges have the same weight one. But in fuzzy graphs, the strength of an edge is a real number in [0, 1] and hence the properties of edges and paths may vary significantly from that of graphs. So it is important to identify and study the nature of edges of fuzzy graphs. In Chap. 2, we have discussed the strength of connectedness between two vertices x and y in a fuzzy graph G. In this chapter, a detailed analysis of the structure of fuzzy graphs based on the strength of connectedness will be made.
Sunil Mathew, John N. Mordeson, Davender S. Malik

### Chapter 4. More on Blocks in Fuzzy Graphs

Abstract
As defined in Chap. 2, a fuzzy graph without fuzzy cutvertices is called a block (nonseparable). Rosenfeld introduced this concept in 1975. In contrast to the classical concept of blocks in graphs, the study of blocks in fuzzy graphs is challenging due to the complexity of cutvertices. Note that cutvertices of a fuzzy graph are those vertices which reduce the strength of connectedness between some pair of vertices on its removal from the fuzzy graph rather than the total disconnection of the fuzzy graph. In this chapter, we concentrate on blocks of fuzzy graphs. This work is from Anjali and Mathew, J Fuzzy Math, 23(4), 907–916 (2015), [28], Anjali and Mathew, J Intell Fuzzy Syst 28, 1659–1665 (2015), [29], Anjali and Mathew, Transitive blocks in fuzzy graphs (2017), [30].
Sunil Mathew, John N. Mordeson, Davender S. Malik

### Chapter 5. More on Connectivity and Distances

Abstract
In this chapter, we discuss more connectivity concepts and distances in fuzzy graphs. The first three sections deal with connectivity and the rest distances. Starting with the first paper of Rosenfeld (Fuzzy sets and their applications, Academic Press, New York, 1975) [154], connectivity was an intense area of research in fuzzy graph theory.
Sunil Mathew, John N. Mordeson, Davender S. Malik

### Chapter 6. Sequences, Saturation, Intervals and Gates in Fuzzy Graphs

Abstract
In this chapter, we discuss three different concepts in fuzzy graphs. The first concept is that of sequences of fuzzy graphs, which allow us to connect a fuzzy graph to a sequence space. Most of the fuzzy graph structures are characterized using different types of sequences. In the second part of this chapter, we discuss saturation in fuzzy graphs and the third part deals with strong intervals and strong gates in fuzzy graphs.
Sunil Mathew, John N. Mordeson, Davender S. Malik

### Chapter 7. Interval-Valued Fuzzy Graphs

Abstract
The results in this chapter are based mostly on the works in Akram (Inf Sci, 181:5548–5564, 2011) [5], Akram and Dudek (Comput Math Appl, 61(2):289–299, 2011) [14], Akram et al. (J Appl Math, 2013, 2013) [19], Akram et al. (Afr math, 2014) [23]. In 1975, Zadeh (Inf Sci, 8:199–249, 1975) [194] introduced the notion of interval-valued fuzzy sets as an extension of fuzzy sets (Zadeh, Inf Control, 8:338–353, 1965, [190]) in which the values of the memberships degrees are intervals in [0, 1] instead of elements in [0, 1]. Interval-valued fuzzy sets provide a more adequate description of uncertainty than traditional fuzzy sets in some cases. It can therefore be important to use interval-valued fuzzy sets in applications, e.g., in fuzzy control.
Sunil Mathew, John N. Mordeson, Davender S. Malik

### Chapter 8. Bipolar Fuzzy Graphs

Abstract
In 1994, Zhang (Proceedings of FUZZ-IEEE, 1998) [195], (Proceedings of IEEE conference, 1994) [71] introduced the concept of bipolar fuzzy sets as a generalization of the notion of Zadeh’s fuzzy sets. A bipolar fuzzy subset of a set is a pair of functions one from the set into the interval [0, 1] and the other into the interval $$[-1,0].$$ In a bipolar fuzzy set, the membership degree 0 of an element can be interpreted that the element is irrelevant to the corresponding property, the membership degree in (0, 1] of an element indicates the intensity that the element satisfies the property, and the membership degree in $$[-1,0)$$ of an element indicates the element does not satisfy the property. Fuzzy and possibilistic formalisms for bipolar information have been proposed in Dubios et al. Inf Pro Man Unc IPMU’04, 2002, [71] because bipolarity exists when dealing with spatial information in image processing or in spatial reasoning applications.
Sunil Mathew, John N. Mordeson, Davender S. Malik

### Backmatter

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