The Maker-Breaker domination game is a positional game played on a graph by two players called Dominator and Staller. The players alternately select a vertex of the graph that has not yet been chosen. Dominator wins if at some point the vertices she has chosen form a dominating set of the graph. Staller wins if Dominator cannot form a dominating set. Deciding if Dominator has a winning strategy has been shown to be a PSPACE-complete problem even when restricted to chordal or bipartite graphs. In this paper, we consider strategies for Dominator based on partitions of the graph into basic subgraphs where Dominator wins as the second player. Using partitions into cycles and edges (also called perfect [1,2]-factors), we show that Dominator always wins in regular graphs and that deciding whether Dominator has a winning strategy as a second player can be computed in polynomial time for outerplanar and block graphs. We then study partitions into subgraphs with two universal vertices, which is equivalent to considering the existence of pairing dominating sets with adjacent pairs. We show that in interval graphs, Dominator wins if and only if such a partition exists. In particular, this implies that deciding whether Dominator has a winning strategy playing second is in NP for interval graphs. We finally provide an algorithm in $n^{k+3}$ for $k$-nested interval graphs (i.e. interval graphs with at most $k$ intervals included one in each other).

翻译：Maker-Breaker 支配游戏是一种在图上的位置性游戏，由两名分别称为支配者（Dominator）和拖延者（Staller）的玩家进行。玩家轮流选择图中一个尚未被选中的顶点。如果支配者所选顶点在某一时刻构成图的一个支配集，则支配者获胜。如果支配者无法形成支配集，则拖延者获胜。已有研究表明，即使限制在弦图或二分图上，判定支配者是否拥有必胜策略也是一个 PSPACE 完全问题。本文考虑基于将图划分为基本子图的支配者策略，在这些子图中支配者作为后手玩家能够获胜。利用将图划分为环和边（也称为完美 [1,2]-因子）的分区，我们证明了支配者在正则图中总是获胜，并且对于外平面图和块图，可以在多项式时间内判定支配者作为后手玩家是否拥有必胜策略。接着，我们研究划分为具有两个通用顶点的子图的分区，这等价于考虑具有相邻顶点对的配对支配集的存在性。我们证明在区间图中，支配者获胜当且仅当存在这样的分区。特别地，这意味着对于区间图，判定支配者作为后手玩家是否拥有必胜策略属于 NP 问题。最后，我们为 $k$-嵌套区间图（即最多有 $k$ 个区间相互包含的区间图）提供了一个 $n^{k+3}$ 时间复杂度的算法。