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Издательство Marcel Dekker, 1998, -500 pp.Within the last twenty-five years, concurrent with the growth of computer science, graph theory has seen explosive growth. Perhaps the fastest growing area within graph theory is the study of domination in graphs.

Cockayne and Hedetniemi's survey paper on domination appeared in 1977 and contained 20 references. This survey paper seems to have set in motion the modern study of dominating sets in graphs. In 1990, Hedetniemi and Laskar edited an issue of Discrete Mathematics devoted entirely to domination. The 1990 bibliography revealed an impressive increase in thirteen years from 20 to approximately 400 references. Seven years later more than 1,000 research papers had been published on dominating sets and related sets in graphs, and the field is steadily growing.

Noting the wide interest, and the need for comprehensive publications in this field of study, we were motivated to produce this book and its companion, Fundamentals of Domination in Graphs. The companion book covers the basics of domination and major research accomplishments on domination in textbook form. It includes the only known comprehensive bibliography on the subject. While writing Fundamentals of Domination in Graphs, it became apparent to us that there was a collection of topics that ought to be covered in greater depth than a textbook would allow. Thus Domination in Graphs: Advanced Topics was conceived.

For this book, we invited leading researchers in domination to contribute chapters. We have been very gratified by the fact that so many of them accepted and have contributed to the project. This book contains 17 chapters, each of which is intended to provide readers with a survey of the known results and to bring them to the forefront of research in a particular aspect of graph domination. These topics are on the frontiers of research and many unsolved problems are presented. Hence it is an important reference work in domination for those interested in research in the area, those wanting to delve more deeply into a specific topic presented here, those interested in a sampling of proof techniques in domination, and those from diverse areas interested in becoming acquainted with the material.

LP-Duality, Complementarity, and Generality of Graphical Subset Parameters.

Dominating Functions in Graphs.

Fractional Domination and Related Parameters.

Majority Domination and Its Generalizations.

Convexity of Extremal Domination-Related Functions of Graphs.

Combinatorial Problems on Chessboards: II.

Domination in Cartesian Products: Vizing's Conjecture.

Algorithms.

Complexity Results.

Domination Parameters of a Graph.

Global Domination.

Distance Domination in Graphs.

Domatic Numbers of Graphs and Their Variants: A Survey.

Domination- Related Parameters.

Topics on Domination in Directed Graphs.

Graphs Critical with Respect to the Domination Number.

Bondage, Insensitivity, and Reinforcement.

Cockayne and Hedetniemi's survey paper on domination appeared in 1977 and contained 20 references. This survey paper seems to have set in motion the modern study of dominating sets in graphs. In 1990, Hedetniemi and Laskar edited an issue of Discrete Mathematics devoted entirely to domination. The 1990 bibliography revealed an impressive increase in thirteen years from 20 to approximately 400 references. Seven years later more than 1,000 research papers had been published on dominating sets and related sets in graphs, and the field is steadily growing.

Noting the wide interest, and the need for comprehensive publications in this field of study, we were motivated to produce this book and its companion, Fundamentals of Domination in Graphs. The companion book covers the basics of domination and major research accomplishments on domination in textbook form. It includes the only known comprehensive bibliography on the subject. While writing Fundamentals of Domination in Graphs, it became apparent to us that there was a collection of topics that ought to be covered in greater depth than a textbook would allow. Thus Domination in Graphs: Advanced Topics was conceived.

For this book, we invited leading researchers in domination to contribute chapters. We have been very gratified by the fact that so many of them accepted and have contributed to the project. This book contains 17 chapters, each of which is intended to provide readers with a survey of the known results and to bring them to the forefront of research in a particular aspect of graph domination. These topics are on the frontiers of research and many unsolved problems are presented. Hence it is an important reference work in domination for those interested in research in the area, those wanting to delve more deeply into a specific topic presented here, those interested in a sampling of proof techniques in domination, and those from diverse areas interested in becoming acquainted with the material.

LP-Duality, Complementarity, and Generality of Graphical Subset Parameters.

Dominating Functions in Graphs.

Fractional Domination and Related Parameters.

Majority Domination and Its Generalizations.

Convexity of Extremal Domination-Related Functions of Graphs.

Combinatorial Problems on Chessboards: II.

Domination in Cartesian Products: Vizing's Conjecture.

Algorithms.

Complexity Results.

Domination Parameters of a Graph.

Global Domination.

Distance Domination in Graphs.

Domatic Numbers of Graphs and Their Variants: A Survey.

Domination- Related Parameters.

Topics on Domination in Directed Graphs.

Graphs Critical with Respect to the Domination Number.

Bondage, Insensitivity, and Reinforcement.

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