On the Tractability of Covering a Graph with 2-Clubs
Covering a graph with cohesive subgraphs is a classical problem in theoretical computer science. In this paper, we prove new complexity results on the Open image in new window problem, a variant recently introduced in the literature which asks to cover the vertices of a graph with a minimum number of 2-clubs (which are induced subgraphs of diameter at most 2). First, we answer an open question on the decision version of Open image in new window that asks if it is possible to cover a graph with at most two 2-clubs, and we prove that it is W-hard when parameterized by the distance to a 2-club. Then, we consider the complexity of Open image in new window on some graph classes. We prove that Open image in new window remains NP-hard on subcubic planar graphs, W-hard on bipartite graphs when parameterized by the number of 2-clubs in a solution and fixed-parameter tractable on graphs having bounded treewidth.
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