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 focused coverage 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.
Show moreThis 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 focused coverage 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.
Show more1. 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.
Teresa W. Haynes has focused her research on domination
in graphs for over 30 years and is perhaps best known for
coauthoring the 1998 book Fundamentals of Domination in
Graphs and the companion volume Domination in Graphs:
Advanced Topics. She has also co-edited 2 volumes in
Springer’s Problem Books in
Mathematics Graph Theory: Favorite Conjectures and
Open Problems. Haynes is also a co-author of
the Springer Briefs in Mathematics From
Domination to Coloring: The Graph Theory of Stephen T.
Hedetniemi. Upon receiving her PhD from the University of
Central Florida in 1988, she joined East Tennessee State
University, where she is currently professor in the Department of
Mathematics and Statistics. Haynes has coauthored more than
200 papers on domination and domination-related concepts, which
introduced some of the most studied concepts in domination, such as
power domination, paired domination, double domination, alliances
and broadcasts in graphs, and stratified
domination.
Stephen T. Hedetniemi is
one of the earliest pioneers of domination in graphs along with E.
J. Cockayne, who together proposed the theory of domination in
graphs, in one of the most cited papers in the field in 1977.
He received his PhD from the University of Michigan in 1966, with
two world-class advisors, graph theorist Frank Harary, and the
pioneer of genetic algorithms and MacArthur Fellowship winner, John
Holland. He coauthored, the first book on domination in
1988 Fundamentals of Domination in Graphs, and co-edited a
second book, Domination in Graphs: Advanced Topics. He also
co-edited 2 volumes in Springer’s Problem Books in
Mathematics Graph Theory: Favorite Conjectures and
Open Problems. Since 1974 he has coauthored more than 300 papers,
180 of which are on domination and domination-related
concepts. Hedetniemi has introduced some of the most-studied
concepts in domination theory, including total domination,
independent domination, irredundance, Roman domination, power
domination, alliances in graphs, signed and minus domination,
fractional domination, domatic numbers, domination in grid graphs
and chessboards, the first domination algorithms, the first
domination NP-completeness results, and the first self-stabilizing
domination algorithms. After leaving the University of
Michigan, he taught computer science at the University of Iowa, and
the University of Virginia, spent a visiting year at the University
of Victoria with E. J. Cockayne, and then became department head of
Computer and information Science at the University of Oregon.
Since 1982 has been at Clemson University, where he served a
five-year term as department head, and served on the Executive
Committee of the Computing Accreditation Commission of ABET, Inc.
He is currently Emeritus Professor of Computer Science in the
School of Computing at Clemson University.
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.
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