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 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 that asks if it is possible to cover a graph with at most two 2-clubs, and we prove that it is W[1]-hard when parameterized by the distance to a 2-club. Then, we consider the complexity of on some graph classes. We prove that remains NP-hard on subcubic planar graphs, W[2]-hard on bipartite graphs when parameterized by the number of 2-clubs in a solution and fixed-parameter tractable on graphs having bounded treewidth.