Structural tools for the Maker-Breaker game. Application to hypergraphs of rank 3: strategies and tractability

In the Maker-Breaker positional game, Maker and Breaker takes turns picking vertices of a hypergraph $H$, and Maker wins if and only if he claims all the vertices of some edge of $H$. This paper mainly details a structural result and an algorithmic result that were presented in conferences by the same authors, in 2020 and 2021 respectively. It also provides a more general framework to study Maker-Breaker games, centered on the notion of danger, which is a subhypergraph representing an urgent threat for Breaker that he must hit with his next pick. Applying this concept in 3-uniform hypergraphs, we exhibit an elementary family of dangers $\mathcal{D}_0$ such that Breaker wins with perfect play if and only if he can hit all dangers from $\mathcal{D}_0$ in each of the first three rounds. This structural criterion has consequences on the algorithmic complexity of deciding which player has a winning strategy on a given hypergraph: this problem, which is known to be PSPACE-complete on 6-uniform hypergraphs, is in polynomial time on hypergraphs of rank 3. This improves on a result by Kutz who showed the same in the linear case, and validates a conjecture by Rahman and Watson. Another corollary of our result is that, if Maker has a winning strategy on a hypergraph of rank 3, then he can ensure to claim an edge in a number of rounds that is logarithmic in the number of vertices.