Upper total domination in claw‐free graphs

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S (other than itself). The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γt(G). We establish bounds on Γt(G) for claw‐free graphs G in terms of the number n of vertices and the minimum degree δ of G. We show that $\Gamma _t(G) \le 2(n+1)/3$ if $\delta \in \{ 1,2\}, \Gamma _t(G) \le 4n/(\delta + 3)$ if $\delta \in \{ 3,4\}$, and $\Gamma _t(G) \le n/2$ if δ ≥ 5. The extremal graphs are characterized. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 148–158, 2003