Trees with equal total domination and game total domination numbers

In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453–1462], where the players Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a total dominating set S of G in which every vertex is totally dominated by a vertex in S. Dominator wishes to minimize the number of vertices chosen, while Staller wishes to maximize it. The game total domination number, γtg(G), (respectively, Staller-start game total domination number, γ ′ tg(G)) of G is the number of vertices chosen when Dominator (respectively, Staller) starts the game and both players play optimally. For general graphs G, sometimes γtg(G) > γ ′ tg(G). We show that if G is a forest with no isolated vertex, then γtg(G) ≤ γ′ tg(G). Using this result, we characterize the trees with equal total domination and game total domination number.