Since its introduction as a Maker-Breaker positional game by Duchêne et al. in 2020, the Maker-Breaker domination game has become one of the most studied positional games on vertices. In this game, two players, Dominator and Staller, alternately claim an unclaimed vertex of a given graph G. If at some point the set of vertices claimed by Dominator is a dominating set, she wins; otherwise, i.e. if Staller manages to isolate a vertex by claiming all its closed neighborhood, Staller wins. Given a graph G and a first player, Dominator or Staller must have a winning strategy. We are interested in the computational complexity of determining which player has such a strategy. This problem is known to be PSPACE-complete on bipartite graphs of bounded degree and split graphs; polynomial on cographs, outerplanar graphs, and block graphs; and in NP for interval graphs. In this paper, we consider the parameterized complexity of this game. We start by considering as a parameter the number of moves of both players. We prove that for the general framework of Maker-Breaker positional games in hypergraphs, determining whether Breaker can claim a transversal of the hypergraph in k moves is W[2]-complete, in contrast to the problem of determining whether Maker can claim all the vertices of a hyperedge in k moves, which is known to be W[1]-complete since 2017. These two hardness results are then applied to the Maker-Breaker domination game, proving that it is W[2]-complete to decide if Dominator can dominate the graph in k moves and W[1]-complete to decide if Staller can isolate a vertex in k moves. Next, we provide FPT algorithms for the Maker-Breaker domination game parameterized by the neighborhood diversity, the modular width, the P4-fewness, the distance to cluster, and the feedback edge number.

翻译：自Duchêne等人于2020年将其作为Maker-Breaker位置游戏提出以来，Maker-Breaker支配游戏已成为顶点上研究最广泛的位置游戏之一。在该游戏中，两名玩家Dominator与Staller轮流占据给定图G中一个未被占据的顶点。若在某一时刻，Dominator占据的顶点集合构成一个支配集，则其获胜；否则，即若Staller通过占据某顶点的全部闭邻域使其孤立，则Staller获胜。给定图G与先手玩家，Dominator或Staller必然存在必胜策略。我们关注于判定哪名玩家拥有此类策略的计算复杂性。已知该问题在有限度二分图与分裂图上是PSPACE完全的；在补图、外平面图与块图上是多项式可解的；在区间图上属于NP。本文中，我们研究该游戏的参数化复杂性。我们首先以双方玩家的移动步数作为参数进行考察。我们证明在超图的Maker-Breaker位置游戏一般框架下，判定Breaker能否在k步内占据超图的一个横截是W[2]完全的，这与判定Maker能否在k步内占据超边所有顶点的问题形成对比——后者自2017年起已知是W[1]完全的。这两个困难性结果随后被应用于Maker-Breaker支配游戏，证明判定Dominator能否在k步内支配该图是W[2]完全的，而判定Staller能否在k步内孤立一个顶点是W[1]完全的。接下来，我们针对以邻域多样性、模宽、P4稀疏度、到簇图的距离及反馈边数作为参数的Maker-Breaker支配游戏，提供了FPT算法。