Methods of Graph Decompositions

Vadim Zverovich · Pavel Skums

ago. de 2024 · Oxford University Press

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In general terms, a graph decomposition is a partition of a graph into parts satisfying some special conditions. Methods of Graph Decompositions discusses some state-of-the-art decomposition methods of graph theory, which are highly instrumental when dealing with a number of fundamental concepts such as unigraphs, isomorphism, reconstruction conjectures, k-dimensional graphs, degree sequences, line graphs and line hypergraphs. The first part of the book explores the algebraic theory of graph decomposition, whose major idea is to define a binary operation that turns the set of graphs or objects derived from graphs into an algebraic semigroup. If an operation and a class of graphs are appropriately chosen, then, just as for integers, each graph has a unique factorization (or canonical decomposition) into a product of prime factors. The unique factorization property makes this type of decomposition especially efficient for problems associated with graph isomorphism, and several such examples are described in the book. Another topic is devoted to Krausz-type decompositions, that is, special coverings of graphs by cliques that are directly associated with representation of graphs as line graphs of hypergraphs. The book discusses various algorithmic and structural results associated with the existence, properties and applications of such decompositions. In particular, it demonstrates how Krausz-type decompositions are directly related to topological dimension, information complexity and self-similarity of graphs, thus allowing to establish links between combinatorics, general topology, information theory and studies of complex systems. The above topics are united by the role played in their development by Professor Regina Tyshkevich, and the book is a tribute to her memory. The book will be ideal for researchers, engineers and specialists, who are interested in fundamental problems of graph theory and proof techniques to tackle them.

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Dr Vadim Zverovich is an Associate Professor of Mathematics and the Head of the Mathematics and Statistics Research Group at the University of the West of England (UWE) in Bristol. An accomplished researcher, he is also a Fellow of the UK Operational Research Society, and was previously a Fellow of the prestigious Alexander von Humboldt Foundation in Germany. In 2016, he was awarded Higher Education Academy fellowship status, and the Faculty of Environment and Technology of the UWE named him its researcher of the year in 2017. His research interests include graph theory and its applications, networks, probabilistic methods, combinatorial optimisation and emergency responses. With 30 years of research experience, he has published many research articles and three books on the above subjects, and established an internationally recognized academic track record in the mathematical sciences covering both theoretical and applied aspects. The most recent book Modern Applications of Graph Theory was published by Oxford University Press in 2021. Dr Pavel Skums is an Associate Professor at the Computer Science and Engineering Department of the University of Connecticut. He is a recipient of the prestigious National Science Foundation CAREER award, GSU Dean of College of Arts and Sciences Early Career Award and US Centers for Disease Control and Prevention Charles C. Shepard Science Award. His research concentrates on graph and network theories and their applications in genomics, epidemiology and immunology. He has published more than 60 papers and book chapters, and has been a guest editor of more than 20 special issues of leading journals.

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