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, $��_t(G)$. 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 distance2 from it. The disjunctive total domination number, $��^d_t(G)$, is the minimum cardinality of such a set. We observe that $��^d_t(G) \le ��_t(G)$. We prove that if $G$ is a connected graph of order$n \ge 8$, then $��^d_t(G) \le 2(n-1)/3$ and we characterize the extremal graphs. It is known that if $G$ is a connected claw-free graph of order$n$, then $��_t(G) \le 2n/3$ 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$n > 10$, then $��^d_t(G) \le 4n/7$ and we characterize the graphs achieving equality in this bound.

23 pages