Positional games were introduced by Hales and Jewett in 1963, and their study became more popular when Erdős and Selfridge showed their connection to Ramsey theory and hypergraph coloring in 1973. Several conventions of these games exist, and the most popular one, Maker-Breaker was proved to be PSPACE-complete by Schaefer in 1978. The study of their complexity then stopped for decades, until 2017 when Bonnet, Jamain, and Saffidine proved that Maker-Breaker is W[1]-complete when parameterized by the number of moves. The study was then intensified when Rahman and Watson improved Schaefer’s result in 2021 by proving that the PSPACE-hardness holds for 6-uniform hypergraphs. More recently, Galliot, Gravier, and Sivignon proved that computing the winner on rank 3 hypergraphs is in P, and Keopke proved that the PSPACE-hardness also holds for 5-uniform hypergraphs. We focus here on the Client-Waiter and the Waiter-Client conventions. Both were proved to be NP-hard by Csernenszky, Martin, and Pluhár in 2011, but neither completeness nor positive results were known. In this paper, we complete the study of these conventions by proving that the former is PSPACE-complete, even restricted to 6-uniform hypergraphs, and by providing an FPT-algorithm for the latter, parameterized by the size of its largest edge. In particular, the winner of Waiter-Client can be computed in polynomial time in rank k hypergraphs for any fixed integer k. Finally, in search of the exact location of the complexity gap in the Client-Waiter convention, we focus on rank 3 hypergraphs. We provide an algorithm that runs in polynomial time with an oracle in NP.

We investigate the parameterized complexity of several problems formalizing cluster identification in graphs. In other words, we ask whether a graph contains a large enough and sufficiently connected subgraph. We study here three relaxations of Clique: s-Club and s-Clique, in which the relaxation is focused on the distances in respectively the cluster and the original graph, and γ-Complete Subgraph in which the relaxation is made on the minimal degree in the cluster. As these three problems are known to be NP-hard, we study here their parameterized complexities. We prove that s-Club and s-Clique are NP-hard even restricted to graphs of degeneracy ≤3 whenever s≥3, and to graphs of degeneracy ≤2 whenever s≥5, which is a strictly stronger result than its W[1]-hardness parameterized by the degeneracy. Concerning γ-Complete Subgraph, we prove that it is W[1]-hard parameterized both by the degeneracy, implying the W[1]-hardness parameterized by the number of vertices in the γ-complete-subgraph, and by the number of elements outside the γ-complete subgraph.

Avoidance games are games in which two players claim vertices of a hypergraph and try to avoid some structures. These games have been studied since the introduction of the game of SIM in 1968, but only few complexity results have been found out about them. In 2001, Slany proved some partial results on Avoider-Avoider games complexity, and in 2017 Bonnet et al. proved that short Avoider-Enforcer games are Co-W[1]-hard. More recently, in 2022, Miltzow and Stojaković proved that these games are NP-hard. As these games correspond to the misère version of the well-known Maker-Breaker games, introduced in 1963 and proven PSPACE-complete in 1978, one could expect these games to be PSPACE-complete too, but the question has remained open since then. Here, we prove here that both Avoider-Avoider and Avoider-Enforcer conventions are PSPACE-complete. Using the PSPACE-hardness of Avoider-Enforcer, we provide in appendix proofs that some particular Avoider-Enforcer games also are.