We introduce the Maker-Breaker domination game, a two player game on a graph. At his turn, the first player, Dominator, select a vertex in order to dominate the graph while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. Both players play alternately without missing their turn. This game is a particular instance of the so-called Maker-Breaker games, that is studied here in a combinatorial context. In this paper, we first prove that deciding the winner of the Maker-Breaker domination game is PSPACE-complete, even for bipartite graphs and split graphs. It is then showed that the problem is polynomial for cographs and trees. In particular, we define a strategy for Dominator that is derived from a variation of the dominating set problem, called the pairing dominating set problem.

中文翻译：

---

Maker–Breaker 统治游戏

我们介绍了 Maker-Breaker 统治游戏，这是一个图形上的两人游戏。轮到他时，第一个玩家 Dominator 选择一个顶点以控制图形，而另一个玩家 Staller 禁止 Dominator 的顶点以防止他达到目标。两名玩家交替玩，不会错过轮到他们。这个游戏是所谓的 Maker-Breaker 游戏的一个特殊实例，这里是在组合环境中研究的。在本文中，我们首先证明决定 Maker-Breaker 统治游戏的获胜者是 PSPACE 完全的，即使对于二部图和分裂图也是如此。然后证明该问题是 cographs 和树的多项式。特别地，我们为支配集问题定义了一种策略，该策略源自支配集问题的变体，称为配对支配集问题。