Domination game critical graphs

17 The domination game is played on a graph G by two players who alternately take 18 turns by choosing a vertex such that in each turn at least one previously undominated 19 vertex is dominated. The game is over when each vertex becomes dominated. One 20 of the players, namely Dominator, wants to finish the game as soon as possible, while 21 the other one wants to delay the end. The number of turns when Dominator starts 22 the game on G and both players play optimally is the graph invariant γg(G), named 23 game domination number. Here we study the γg-critical graphs which are critical 24 with respect to vertex predomination. Besides proving some general properties, we 25 characterize γg-critical graphs with γg = 2 and with γg = 3, moreover for each n we 26 identify the (infinite) class of all γg-critical ones among the nth powers C n N of cycles. 27 Along the way we determine γg(C n N ) for all n and N . Results of a computer search 28 for γg-critical trees are presented and several problems and research directions are also 29 listed. 30