Domination in Graphs

• Nawarat Ananchuen
• Watcharaphong Ananchuen
• Michael D. Plummer
Chapter

Abstract

A set of vertices S in a graph G dominates G if every vertex in G is either in S or adjacent to a vertex in S. The size of any smallest dominating set is called the domination number of G. Two variants on this concept that have attracted recent interest are total domination and connected domination. A set of vertices S is a total dominating set if every vertex in the graph is adjacent to a vertex of S and S is a connected dominating set if it is dominating and, in addition, induces a connected subgraph. The size of any smallest total dominating set in G is called the total domination number of G and the size of a smallest connected dominating set is the connected domination number of G. These simple, yet wide-ranging, graph-theoretic concepts have a multitude of real-world applications. There are already in print several surveys of results on domination; therefore, in this chapter we adopt a slightly different approach. We begin by surveying results on bounding the three domination numbers. We then focus on criticality concepts for domination. The two types of criticality most widely studied to date are graphs for which the domination number decreases upon the addition of any missing edge and the graphs for which the domination number decreases upon the deletion of any vertex. Recently, there has been increased activity in the study of these critical concepts and we survey these new results, focusing especially upon matching in critical graphs.

Keywords

Domination Total domination Connected domination Edge critical Vertex critical Matching

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