Total domination in claw-free graphs with minimum degree 2

A set S of vertices in a graph G is a total dominating set (TDS) of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a TDS of G is the total domination number of G, denoted by γ_t (G). A graph is claw-free if it does not contain K_{1, 3} as an induced subgraph. It is known [M.A. Henning, Graphs with large total domination number, J. Graph Theory 35(1) (2000) 21-45] that if G is a connected graph of order n with minimum degree at least two and G ∉ { C₃, C₅, C₆, C₁₀ }, then γ_t (G) ≤ 4 n / 7. In this paper, we show that this upper bound can be improved if G is restricted to be a claw-free graph. We show that every connected claw-free graph G of order n and minimum degree at least two satisfies γ_t (G) ≤ (n + 2) / 2 and we characterize those graphs for which γ_t (G) = ⌊ (n + 2) / 2 ⌋.