Total domination game on ladder graphs

The total domination game is played on a simple graph G by two players, named Dominator and Staller. They alternately select a vertex of G; each chosen vertex totally dominates its neighbors. In this game, each chosen vertex must totally dominate at least one new vertex not totally dominated before. The game ends when all vertices in G are totally dominated. Dominator's goal is to finish the game as soon as possible, and Staller's goal is to prolong it as much as possible. The game total domination number is the number of chosen vertices when both players play optimally. There are two types of such number, one where Dominator starts the game and another where Staller starts the game. In this paper, we determine the game total domination numbers of the ladders, the circular ladders and the Möbius ladders