On graphs with small game domination number
Résumé
The domination game is played on a graph G by Dominator and Staller. The game domination number γg(G) of G is the number of moves played when Dominator starts and both players play optimally. Similarly, γ′g(G) is the number of moves played when Staller starts. 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