On the parameterized complexity of the Maker-Breaker domination game

TL;DR

The paper analyzes the parameterized complexity of the Maker-Breaker domination game on graphs, relating the move-based decision problems to classical and parameterized complexity classes. It proves a dichotomy: Short problems for Maker-type winners sit at $[1]$-completeness, while Breaker-oriented and global-win variants require $[2]$-complete frameworks, including a general result that Breaker can win in $k$ moves is $[2]$-complete. Beyond move-count, it develops fixed-parameter algorithms and kernels for structural graph parameters such as neighborhood diversity, modular width, $P_4$-fewness, distance to cluster, and the feedback edge number, leveraging module-replacement lemmas and modular-decomposition techniques. These results advance understanding of fixed-parameter tractability in Maker-Breaker games and provide practical kernels and algorithms for a broad class of graphs. The work highlights both the role of global winning conditions in Breaker’s complexity and the utility of modular and decomposition-based reductions for algorithmic game analysis.

Abstract

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.

Figures (9)

• Figure 1: Summary of our results on structural parameters. If a parameter is bounded for a class of graphs, it is bounded for its descendants in the diagram.
• Figure 2: The different outcomes in the MB Dom Game
• Figure 3: Split on the vertex $v_0$ using Lemma \ref{['removing vertices Dominator']}. The two graphs have the same outcome.
• Figure 4: The reduction in Theorem \ref{['thm short dom game by staller']}, with $k= 2$
• Figure 5: The reduction in Theorem \ref{['shortDD W2 hard']}

Theorems & Definitions (49)

• Theorem 2.2: makerbreakerOijid2025
• Proposition 2.3
• proof
• Lemma 2.4: makerbreaker
• Lemma 2.5: makerbreaker
• Lemma 2.8
• proof
• Lemma 2.9
• proof
• Lemma 2.10: makerbreaker