🤖 AI Summary
This work addresses the computational complexity of determining the winner in the unbiased Maker-Breaker triangle game on general graphs. Introducing, for the first time, an edge-triangle incidence graph model combined with structural characterizations and monotone strategies, the paper establishes necessary and sufficient conditions for Maker’s victory. Leveraging this framework, it designs polynomial-time algorithms that reduce the decision complexity from $O(n^{16})$ to $O(n^7)$ for arbitrary graphs; achieve $O(n^{\omega+1})$ when the graph contains a $K_4$ and its incidence graph is connected; and attain $O(n^3)$ for $K_4$-free cactus-like incidence graphs. Furthermore, the study presents a linear-time reduction from triangle detection to game outcome determination, significantly enhancing computational efficiency and broadening the class of applicable graphs.
📝 Abstract
In this paper, we present new polynomial-time algorithms for determining the winner of the unbiased triangle game played on the edge set of general graphs. To that end, we propose to view the game through the edge-triangle incidence graph instead of the standard hypergraph model. We identify a necessary and sufficient winning condition for Maker in terms of the edge-triangle incidence graph and show that winning strategies achieving this condition as fast as possible play monotonically in the sense that they only consider monotonically decreasing connected subgraphs of the edge-triangle incidence graph.
We give three different algorithms for different classes of graphs. For general graphs $G$, the outcome of the unbiased triangle game can be decided in time $\mathcal{O}(n+m^{3.5})=\mathcal{O}(n^7)$. This significantly improves on the $\mathcal{O}(n^{16})$ algorithm implied by the work of Galliot, Gravier and Sivignon (arXiv 2022). For graphs $G$ which contain $K_4$, the complete graph on four vertices, as a subgraph and whose edge-triangle incidence graph is connected, the winner can be decided in time $\mathcal{O}(n+\min\{n^{ω+1},m^2\})=\mathcal{O}(n^{ω+1})$, where $ω<2.372$ is the exponent of matrix multiplication (Alman et. al., SODA 2025). For graphs $G$ whose edge-triangle incidence graph is a cactus graph, i.e. all its cycles are edge-disjoint, the winner can be decided in time $\mathcal{O}(n+m^{1.5})=\mathcal{O}(n^3)$. Such $G$ are $K_4$-free. The algorithms for the special cases are based on novel structural characterizations of Maker's win for each graph class. We also give a linear time reduction from triangle detection to deciding the unbiased triangle game.