Domination Game Critical Graphs

Abstract The domination game is played on a graph G by two players who alternately take turns by choosing a vertex such that in each turn at least one previously undominated vertex is dominated. The game is over when each vertex becomes dominated. One of the players, namely Dominator, wants to finish the game as soon as possible, while the other one wants to delay the end. The number of turns when Dominator starts the game on G and both players play optimally is the graph invariant γg(G), named the game domination number. Here we study the γg-critical graphs which are critical with respect to vertex predomination. Besides proving some general properties, we characterize γg-critical graphs with γg = 2 and with γg = 3, moreover for each n we identify the (infinite) class of all γg-critical ones among the nth powers CnN of cycles. Along the way we determine γg(CnN) for all n and N. Results of a computer search for γg-critical trees are presented and several problems and research directions are also listed.