On the Game Total Domination Number

Abstract

The total domination game is a two-person competitive optimization game, where the players, Dominator and Staller, alternately select vertices of an isolate-free graph G. Each vertex chosen must strictly increase the number of vertices totally dominated. This process eventually produces a total dominating set of G. Dominator wishes to minimize the number of vertices chosen in the game, while Staller wishes to maximize it. The game total domination number of G, $$\gamma _{\mathrm{tg}}(G)$$ , is the number of vertices chosen when Dominator starts the game and both players play optimally. Recently, Henning, Klavžar, and Rall proved that $$\gamma _{\mathrm{tg}}(G) \le \frac{4}{5}n$$ holds for every graph G which is given on n vertices such that every component of it is of order at least 3; they also conjectured that the sharp upper bound would be $$\frac{3}{4}n$$ . Here, we prove that $$\gamma _{\mathrm{tg}}(G) \le \frac{11}{14}n$$ holds for every G which contains no isolated vertices or isolated edges.