4-Total domination game critical graphs

The total domination game is played on a simple graph [Formula: see text] with no isolated vertices by two players, Dominator and Staller, who alternate choosing a vertex in [Formula: see text]. Each chosen vertex totally dominates its neighbors. In this game, each chosen vertex must totally dominates at least one new vertex not totally dominated before. The game ends when all vertices in [Formula: see text] 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, denoted by [Formula: see text] when Dominator starts the game and denoted by [Formula: see text] when Staller starts the game. If a vertex [Formula: see text] in [Formula: see text] is declared to be already totally dominated, then we denote this graph by [Formula: see text]. A total domination game critical graph is a graph [Formula: see text] for which [Formula: see text] holds for every vertex [Formula: see text] in [Formula: see text]. If [Formula: see text], then [Formula: see text] is called [Formula: see text]-[Formula: see text]-critical. In this work, we characterize some 4-[Formula: see text]-critical graphs.