Disjunctive total domination in graphs

Abstract

Let $$G$$ be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, $$\gamma _t(G)$$ . A set $$S$$ of vertices in $$G$$ is a disjunctive total dominating set of $$G$$ if every vertex is adjacent to a vertex of $$S$$ or has at least two vertices in $$S$$ at distance $$2$$ from it. The disjunctive total domination number, $$\gamma ^d_t(G)$$ , is the minimum cardinality of such a set. We observe that $$\gamma ^d_t(G) \le \gamma _t(G)$$ . We prove that if $$G$$ is a connected graph of order $$n \ge 8$$ , then $$\gamma ^d_t(G) \le 2(n-1)/3$$ and we characterize the extremal graphs. It is known that if $$G$$ is a connected claw-free graph of order $$n$$ , then $$\gamma _t(G) \le 2n/3$$ and this upper bound is tight for arbitrarily large $$n$$ . We show this upper bound can be improved significantly for the disjunctive total domination number. We show that if $$G$$ is a connected claw-free graph of order $$n > 14$$ , then $$\gamma ^d_t(G) \le 4n/7$$ and we characterize the graphs achieving equality in this bound.