On structural parameterizations for the 2-club problem

The NP-hard 2-Club problem is, given an undirected graph G=(V,E)G=(V,E) and  ℓ∈Nℓ∈N, to decide whether there is a vertex set  S⊆VS⊆V of size at least  ℓℓ such that the induced subgraph  G[S]G[S] has diameter at most two. We make progress towards a systematic classification of the complexity of 2-Club with respect to a hierarchy of prominent structural graph parameters. First, we present the following tight NP-hardness results: 2-Club is NP-hard on graphs that become bipartite by deleting one vertex, on graphs that can be covered by three cliques, and on graphs with domination number two and diameter three. Then, we consider the parameter h-index of the input graph. The study of this parameter is motivated by real-world instances and the fact that 2-Club is fixed-parameter tractable when parameterized by the larger parameter maximum degree. We present an algorithm that solves 2-Club in  |V|f(k)|V|f(k) time with  kk being the h-index   of  GG. By showing W[1]-hardness for this parameter, we provide evidence that the above algorithm cannot be improved to a fixed-parameter algorithm. Furthermore, the reduction used for this hardness result can be modified to show that 2-Club is NP-hard if the input graph has constant degeneracy. Finally, we show that 2-Club is fixed-parameter tractable when parameterized by distance to cographs.