Domination Game: A proof of the $3/5$-Conjecture for Graphs with Minimum Degree at Least Two

Abstract

In the domination game on a graph $G$, the players Dominator and Staller alternately select vertices of $G$. Each vertex chosen must strictly increase the number of vertices dominated. This process eventually produces a dominating set of $G$; Dominator aims to minimize the size of this set, while Staller aims to maximize it. The size of the dominating set produced under optimal play is the game domination number of $G$, denoted by $\gamma_g (G)$. In this paper, we prove that $\gamma_g(G) \le 2n/3$ for every $n$-vertex isolate-free graph $G$. When $G$ has minimum degree at least $2$, we prove the stronger bound $\gamma_g(G) \le 3n/5$; this resolves a special case of a conjecture due to Kinnersley, West, and Zamani [SIAM J. Discrete Math., 27 (2013), pp. 2090--2107]. Finally, we prove that if $G$ is an $n$-vertex isolate-free graph with $\ell$ vertices of degree 1, then $\gamma_g(G) \le 3n/5 + \left \lceil \ell/2 \right \rceil + 1$; in the course of establishing this result, we answer a question of Brešar et al. [Discrete Math., 330 (2014), pp. 1--10].