Extremal Problems for Game Domination Number
William B. Kinnersley, Douglas B. West, Reza Zamani · SIAM Journal on Discrete Mathematics · 2013
In the domination game on a graph $G$, two players called Dominator and Staller alternately select vertices of $G$. Each vertex chosen must strictly increase the number of vertices dominated; the game ends when the chosen set becomes a dominating set of $G$. Dominator aims to minimize the size of the resulting dominating set, while Staller aims to maximize it. When both players play optimally, the size of the dominating set produced is the game domination number of $G$, denoted by $\gamma_g(G)$ when Dominator plays first and by $\gamma_g^\prime(G)$ when Staller plays first. We prove that $\gamma_g(G) \le 7n/11$ when $G$ is an isolate-free $n$-vertex forest and that $\gamma_g(G) \le \left\lceil7n/10\right\rceil$ for any isolate-free $n$-vertex graph. In both cases we conjecture that $\gamma_g(G) \le 3n/5$ and prove it when $G$ is a forest of nontrivial caterpillars. We also resolve conjectures of Brešar, Klavžar, and Rall by showing that always $\gamma_g^\prime(G)\le\gamma_g(G)+1$, that for $k\ge2$ there are graphs $G$ satisfying $\gamma_g(G) = 2k$ and $\gamma_g^\prime(G) = 2k-1$, and that $\gamma_g^\prime(G) \ge \gamma_g(G)$ when $G$ is a forest. Our results follow from fundamental lemmas about the domination game that simplify its analysis and may be useful in future research.