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On the Parameterized Complexity of Relaxations of Clique

by   Ambroise Baril, et al.

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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