Hardness and tractability of the γ-Complete Subgraph problem

Finding cohesive subgraphs is a fundamental problem for the analysis of graphs. Clique is a classical model of cohesive subgraph, but several alternative definitions have been given in the literature. In this paper we consider the γ-complete graph model, which is based on relaxing the degree constraint of the clique model, that is G[S] is a γ-complete graph, with γ∈(0,1], if and only if every vertex of S has degree at least γ(|S|−1) in G[S]. In this contribution, we investigate the complexity of the problem that asks for the γ-complete subgraph of maximum order (that is, maximum number of vertices). We show that the problem is W[1]-hard for [Formula presented] when the parameter is order of the subgraph. Moreover, we show that the problem is fixed-parameter tractable when parameterized by the h-index of the input graph, thus solving an open question in the literature.