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This monograph is designed to be an in-depth introduction to domination in graphs. It focuses on three core concepts: domination, total domination, and independent domination. It contains major results on these foundational domination numbers, including a wide variety of in-depth proofs of selected results providing the reader with a toolbox of proof techniques used in domination theory. Additionally, the book is intended as an invaluable reference resource for a variety of readerships, namely, established researchers in the field of domination who want an updated, comprehensive coverage of domination theory; next, researchers in graph theory who wish to become acquainted with newer topics in domination, along with major developments in the field and some of the proof techniques used; and, graduate students with interests in graph theory, who might find the theory and many real-world applications of domination of interest for masters and doctoral thesis topics. The focusedcoverage also provides a good basis for seminars in domination theory or domination algorithms and complexity.
The authors set out to provide the community with an updated and comprehensive treatment on the major topics in domination in graphs. And by Jove, they've done it! In recent years, the authors have curated and published two contributed volumes: Topics in Domination in Graphs , © 2020 and Structures of Domination in Graphs , © 2021. This book rounds out the coverage entirely. The reader is assumed to be acquainted with the basic concepts of graph theory and has had some exposure to graph theory at an introductory level. As graph theory terminology sometimes varies, a glossary of terms and notation is provided at the end of the book.
"This monograph is a precious asset to the domination theory family and, indeed, the entire graph theory community. It showcases the beauty of the field and delivers a powerful message that there are many problems worth studying that are rich and deep enough to build a career on." (Christina M. Mynhardt, Mathematical Reviews, July, 2024)
"One remarkable aspect of this book is its readability. Despite the complex subject matter, the authors have succeeded in making the material accessible to a wide range of readers. . The book's structure is well-organized . . It covers a wide range of topics, from historical origins and fundamental concepts to complexity and algorithms, upper bounds, probabilistic bounds, and much more. Each chapter offers a comprehensive exploration of its respective subject matter, complete with proofs and examples . ." (Aleksandra Tepeh, zbMATH 1526.05002, 2024)
Michael A. Henning has devoted much of his research interests to the field of domination theory in graphs. He has been both plenary and invited speakers at several international conferences and is a prolific researcher having published over 460 papers to date in international mathematics journals. Henning was born and schooled in South Africa having obtained his PhD at the University of Natal in April 1989. In January 1989, he started his academic career as a lecturer at the University of Zululand, before accepting a lectureship in mathematics at the former University of Natal in January 1991. In January 2000, he was appointed a full professor at the University of Natal, which later merged with the University of Durban-Westville to form the University of KwaZulu-Natal in January 2004. After spending almost 20years at the University of KwaZulu-Natal and one of its predecessors, the University of Natal, Michael moved to the University of Johannesburg in May 2010 as a research professor. Most recently he co-authored a unique and stunning textbook in the Springer Optimization and its Applications series titled Graph and Network Theory . He co-authored a Springer Briefs in Mathematics From Domination to Coloring: The Graph Theory of Stephen T. Hedetniemi and co-authored the Springer Monographs in Mathematics book Total Domination in Graphs and in 2020, he co-authored Springer's Developments in Mathematics book Transversals in Linear Uniform Hypergraphs .
1. Introduction.- 2. Historic background.- 3. Domination Fundamentals.- 4. Bounds in terms of order and size, and probability.- 5. Bounds in terms of degree.- 6. Bounds with girth and diameter conditions.- 7. Bounds in terms of forbidden subgraphs.- 8. Domination in graph families : Trees.- 9. Domination in graph families: Claw-free graphs.- 10. Domination in regular graphs including Cubic graphs.- 11. Domination in graph families: Planar graph.- 12. Domination in graph families: Chordal, bipartite, interval, etc.- 13. Domination in grid graphs and graph products.- 14. Progress on Vizing's Conjecture.- 15. Sums and Products (Nordhaus-Gaddum).- 16. Domination Games.- 17. Criticality.- 18. Complexity and Algorithms.- 19. The Upper Domination Number.- 20. Domatic Numbers (for lower and upper gamma) and other dominating partitions, including the newly introduced Upper Domatic Number.- 21. Concluding Remarks, Conjectures, and Open Problems.
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