The domination game played on a graph $G$ consists of two players, Dominator and Staller, who alternate taking turns choosing a vertex from $G$ such that whenever a vertex is chosen by either player, at least one additional vertex is dominated. Dominator wishes to dominate the graph in as few steps as possible, and Staller wishes to delay the process as much as possible. The game domination number $\gamma_g(G)$ (resp., $\gamma_g'(G)$) is the number of vertices chosen when Dominator (resp., Staller) starts the game. An imagination strategy is developed as a general tool for proving results on the domination game. We show that for any graph $G$, $\gamma(G)\leq\gamma_g(G)\leq2\gamma(G)-1$, and that all possible values can be realized. It is proved that for any graph $G$, $\gamma_g(G)-1\leq\gamma'_g(G)\leq\gamma_g(G)+2$, and that most of the possibilities for mutual values of $\gamma_g(G)$ and $\gamma_g'(G)$ can be realized. A connection with Vizing's conjecture is established, and a lower bound on the game domination number of an arbitrary Cartesian product is proved. Several problems and conjectures are also stated.

/pdf/domination-game-and-an-imagination-strategy-17zzntl9so.pdf

Domination Game and an Imagination Strategy

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 Bresar, Klavžar, and Rall by showing that always $\gamma_g^\prime(G)\le\gamma_g(G)+1$, that for $k\ge2$ there a...

/pdf/extremal-problems-for-game-domination-number-2ylc54tizm.pdf

Extremal Problems for Game Domination Number

Many graph theoretic algorithms rely on an initial ordering of the vertices of the graph which has some special properties. We discuss new ways to measure the quality of such orders, give methods for constructing high quality orders, and provide applications for these orders. While our main motivation is the study of game chromatic number, there have been other applications of these ideas and we expect there will be more.

Thomas Dinski

Papers

A bound for the game chromatic number of graphs