On graphs with small game domination number

The domination game is played on a graph G by Dominator and Staller. The two players are taking turns choosing a vertex from G such that at least one previously undominated vertex becomes dominated; the game ends when no move is possible. The game is called D-game when Dominator starts it, and S-game otherwise. Dominator wants to finish the game as fast as possible, while Staller wants to prolong it as much as possible. The game domination number γg(G) of G is the number of moves played in D-game when both players play optimally. Similarly, γ′ g(G) is the number of moves played in S-game. Graphs G with γg(G) = 2, graphs with γ ′ g(G) = 2, as well as graphs extremal with respect to the diameter among these graphs are characterized. In particular, γ′ g(G) = 2 and diam(G) = 3 hold for a graph G if and only if G is a so-called gamburger. Graphs G with γg(G) = 3 and diam(G) = 6, as well as graphs G with γ′ g(G) = 3 and diam(G) = 5 are also characterized. The latter can be described as the so-called double-gamburgers.