This book explores the fundamentals of total domination in graphs, the interplay with transversals in hypergraphs and the association with diameter-2-critical graphs. Includes several proofs, and a toolbox of proof techniques for attacking open problems.

Total Domination in Graphs gives a clear understanding of this topic to any interested reader who has a modest background in graph theory. This book provides and explores the fundamentals of total domination in graphs. Some of the topics featured include the interplay between total domination in graphs and transversals in hypergraphs, and the association with total domination in graphs and diameter-2-critical graphs. Several proofs are included in this text which enables readers to acquaint themselves with a toolbox of proof techniques and ideas with which to attack open problems in the field. This work is an excellent resource for students interested in beginning their research in this field. Additionally, established researchers will find the book valuable to have as it contains the latest developments and open problems.

Provides a comprehensive treatment on total domination in graphs Includes a chapter on open questions and conjectures is presented for researchers in the field Features topics that include the interaction between total domination in graphs and transversals in hypergraphs, the association with total domination in graphs and diameter-2-critical graphs? Investigates upper bounds on the total domination number of a planar graph of small diameter Includes supplementary material: sn.pub/extras

Autorentext

Michael Henning is a world-leader in domination theory in graphs. He has been a plenary and invited speaker at several international conferences and is a prolific researcher having published over 300 papers to date in international mathematics journals. Michael 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 Michael 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 20 years 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. Anders Yeo is a world-leader in several areas within mathematics and computer science, including total domination in graphs and transversal in hypergraphs, digraphs, and in Fixed Parameter Tractability. He has been a plenary and invited speaker at several international conferences and is a prolific researcher having published over 130 papers to date in international mathematics journals. Anders was born in Australia and schooled in Denmark having obtained his PhD at Odense University in December 1997. Thereafter he pursued postdoctoral studies in the Department of Mathematics and Statistics at the University of Victoria in Canada for one year and postdoctoral studies in the Computer Science Department at the University of Aarhus in Denmark for a further two years. In September 2001 he began as a lecturer at Royal Holloway, University of London and was very rapidly promoted to Reader in March 2003 where he worked until January 2012. Anders is currently employed by the University of Johannesburgin a research capacity.

Inhalt

1. Introduction.- 2. Properties of Total Dominating Sets and General Bounds.- 3. Complexity and Algorithmic Results.- 4.Total Domination in Trees.- 5.Total Domination and Minimum Degree.- 6. Total Domination in Planar Graphs.- 7. Total Domination and Forbidden Cycles.- 8. Relating the Size and Total Domination Number.- 9. Total Domination in Claw-Free Graphs.- 10. Total Domination Number versus Matching Number.- 11. Total Domination Critical Graphs.- 12. Total Domination and Graph Products.- 13. Graphs with Disjoint Total Dominating Sets.- 14. Total Domination in Graphs with Diameter Two.- 15. Nordhaus-Gaddum Bounds for Total Domination.- 16. Upper Total Domination.- 17.Variations of Total Domination.- 18. Conjectures and Open Problems.- Index.