On the parameterized complexity of non-hereditary relaxations of clique

Abstract

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.