On graphs with small game domination number

The domination game is played on a graph G by Dominator and Staller. The game domination number ?(G) of G is the number of moves played when Dominator starts and both players play optimally. Similarly, ?g (G) is the number of moves played when Staller starts. Graphs G with ?(G) = 2, graphs with ?g(G) = 2, as well as graphs extremal with respect to the diameter among these graphs are characterized. In particular, ?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 ?(G) = 3 and diam(G) = 6, as well as graphs G with ?g(G) = 3 and diam(G) = 5 are also characterized.