Disjunctive total domination in graphs

Let $$G$$ 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, $$\gamma _t(G)$$ γ t ( G ) . A set $$S$$ S of vertices in $$G$$ G is a disjunctive total dominating set of $$G$$ G if every vertex is adjacent to a vertex of $$S$$ S or has at least two vertices in $$S$$ S at distance $$2$$ 2 from it. The disjunctive total domination number, $$\gamma ^d_t(G)$$ γ t d ( G ) , is the minimum cardinality of such a set. We observe that $$\gamma ^d_t(G) \le \gamma _t(G)$$ γ t d ( G ) ≤ γ t ( G ) . We prove that if $$G$$ G is a connected graph of order $$n \ge 8$$ n ≥ 8 , then $$\gamma ^d_t(G) \le 2(n-1)/3$$ γ t d ( G ) ≤ 2 ( n - 1 ) / 3 and we characterize the extremal graphs. It is known that if $$G$$ G is a connected claw-free graph of order $$n$$ n , then $$\gamma _t(G) \le 2n/3$$ γ t ( G ) ≤ 2 n / 3 and this upper bound is tight for arbitrarily large $$n$$ n . We show this upper bound can be improved significantly for the disjunctive total domination number. We show that if $$G$$ G is a connected claw-free graph of order $$n > 14$$ n > 14 , then $$\gamma ^d_t(G) \le 4n/7$$ γ t d ( G ) ≤ 4 n / 7 and we characterize the graphs achieving equality in this bound.