In this paper, we continue the study of the total domination game in graphs introduced in Henning et al. (2015), where the players Dominator and Staller alternately select vertices of   G     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     S        of   G     G        in which every vertex is totally dominated by a vertex in   S     S       . Dominator wishes to minimize the number of vertices chosen, while Staller wishes to maximize it. The game total domination number,              γ     tg      (  G  )        , (respectively, Staller-start game total domination number,              γ     tg     ′     (  G  )        ) of   G     G        is the number of vertices chosen when Dominator (respectively, Staller) starts the game and both players play optimally. For general graphs   G     G       , sometimes              γ     tg      (  G  )   >    γ     tg     ′     (  G  )        . We show that if   G     G        is a forest with no isolated vertex, then              γ     tg      (  G  )   ≤    γ     tg     ′     (  G  )        . Using this result, we characterize the trees with equal total domination and game total domination number.
['Michael A. Henning', 'Douglas F. Rall']
Trees with Equal Total Domination and Game Total Domination Numbers