On the Parameterized Complexity of Relaxations of Clique
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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. We also obtain that these problems are solvable in polynomial time on graphs of degeneracy 1. Concerning γ-COMPLETE SUBGRAPH, we prove that it is W[1]-hard parameterized by both the degeneracy, which implies the W[1]-hardness parameterized by the number of vertices in the γ-complete-subgraph, and the number of elements outside the γ-complete subgraph.
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