On Total Domination in Graphs

LetG = (V,E) be a finite, simple, undirected graph. A set S ⊆ V is called a total dominating set if every vertex of V is adjacent to some vertex of S. Interest in total domination began when the concept was introduced by Cockayne, Dawes, and Hedetniemi [6] in 1980. In 1998, two books on the subject appeared ([11] and [12]), followed by a survey of more recent results in 2009 [15]. The total domination number of a graph G, denoted γt(G) is the cardinality of a smallest total dominating set. The problem of computing γt(G) for a given graph was shown to be NP-complete by Pfaff et al. [20]. However, a linear algorithm exists for trees [18]. The overall complexity of calculating γt(G) is the primary motivation for discovering bounds, preferably in terms of easily computable invariants. For this project, DeLaViña’s conjecturing program Graffiti.pc was used to investigate lower and upper bounds for the total domination number of a graph. We survey a few known results on total domination and discuss the conjectures of Graffiti.pc that were resolved during the project. Proofs are supplied for true conjectures and counterexamples are given for false conjectures. In some cases, smallest counterexamples are provided.