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 ${\gamma }_g(G)\le \lceil \frac{n(G)}{2}\rceil -\lfloor \frac{n(G)}{11}\rfloor$. The bound is attained: if $\mathrm{diam}(G)=2$ and $n(G)\le 10$, then ${\gamma }_g(G)=\lceil \frac{n(G)}{2}\rceil$ 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.