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, ${\gamma }_{\mathrm{t}\mathrm{g}}(G)$, (respectively, Staller-start game total domination number, ${\gamma }_{\mathrm{t}\mathrm{g}}^{\mathrm{\prime }}(G)$) 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 $\gamma_{\rm tg}(G) > \gamma_{\rm tg}'(G)$\mathrm{.}Weshowthatif$G$ is a forest with no isolated vertex, then ${\gamma }_{\mathrm{t}\mathrm{g}}(G)\le {\gamma }_{\mathrm{t}\mathrm{g}}^{\mathrm{\prime }}(G)$. Using this result, we characterize the trees with equal total domination and game total domination number.