Preface

Lotfi Zadeh introduced the concept of a fuzzy subset of a set in 1965 as a way to represent uncertainty. His ideas have motivated the interest of researchers worldwide. One such researcher was Azriel Rosenfeld. He was one of the fathers of fuzzy graph theory. His development of the concept of a fuzzy graph provides the motivation of this book and the research it contains. The book deals with current ideas in fuzzy graphs. It is not an attempt to provide an exhaustive study. There are individual topics in fuzzy graphs that would provide enough material for an entire book in them. Still it covers most of the major developments in fuzzy graph theory during the period 1975–2017. The book should be of interest to research mathematicians, computer scientists, and social scientists. It is the first volume of a two volume set. The second volume focuses on the application of fuzzy graph theory to the problem of human trafficking. Some of the material in this book has appeared in [127]. We include it here since the development of the book rests on it. We provide in Chap. 1 only the very basics of fuzzy set theory needed to understand the book. We assume the reader is familiar with basic notions of mathematics including set theory. Since this book is designed primarily for researchers with a knowledge of fuzzy set theory, we only provide a few concepts from fuzzy sets and relations mainly to set forth our notation to be used in the book. In Chap. 2, we present basic concepts of fuzzy graphs which are needed later in the chapter and in the remainder of the book. For example, we introduce and present basic results on paths, connectedness, forests, trees, and fuzzy cutsets. Other basic concepts include bridges, cutsets, and blocks. We examine the connection between cycles and fuzzy trees. We present deeper results on blocks and in fact give a characterization of blocks in fuzzy graphs. We examine special types of cycles such as strong cycles and locamin cycles. We then present results on important