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 Γ_t(G). We establish bounds on Γ_t(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) ≤2(n+1)/3 if δ ∈ {1,2}, Γ_t(G)≤ 4n/(δ + 3) if δ ∈ {3,4}, and Γ_t(G)≤ n/2 if δ ≥ 5. The extremal graphs are characterized.