2025/02/27 by Finn Orson Koepke, Koepke, Finn Orson · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Limits and Structures in Graph Theory #Complexity and Algorithms in Graphs
paper · pdf · doi:10.48550/arxiv.2502.20271
Let (X, F) be a hypergraph. The Maker-Breaker game on (X, F) is a combinatorial game between two players, Maker and Breaker. Beginning with Maker, the players take turns claiming vertices from X that have not yet been claimed. Maker wins if she manages to claim all vertices of some hyperedge F ∈ F. Breaker wins if he claims at least one vertex in every hyperedge. M. L. Rahman and Thomas Watson proved in 2021 that, even when only Maker-Breaker games on 6-uniform hypergraphs are considered, the decision problem of determining which player has a winning strategy is PSPACE-complete. They also showed that the problem is NL-hard when considering hypergraphs of rank 5. In this paper, we improve the latter result by showing that deciding who wins Maker-Breaker games on 5-uniform hypergraphs is still a PSPACE-complete problem. We achieve this by polynomial transformation from the problem of solving the generalized geography game on bipartite digraphs with vertex degrees 3 or less, which is known to be PSPACE-complete.