The domination game played on diameter 2 graphs

Abstract

Let $$\gamma _g(G)$$ be the game domination number of a graph G. It is proved that if $$\mathrm{diam}(G) = 2$$ , then $$\gamma _g(G) \le \left\lceil \frac{n(G)}{2} \right\rceil - \left\lfloor \frac{n(G)}{11}\right\rfloor $$ . The bound is attained: if $$\mathrm{diam}(G) = 2$$ and $$n(G) \le 10$$ , then $$\gamma _g(G) = \left\lceil \frac{n(G)}{2} \right\rceil $$ if and only if G is one of seven sporadic graphs with $$n(G)\le 6$$ or the Petersen graph, and there are exactly ten graphs of diameter 2 and order 11 that attain the bound.