Abstract:
Given an undirected graph G = (V,E), an s-club is a vertex subset S ⊆ V (G) such that G[S] has diameter at most s. Formally, an s-Club problem asks if the input graph has an s-club with at least k vertices. There have been plethora of works on s-Club problem with fixed value of s = 2 as well as with vertex and edge triangle constraints. Vertex-ℓ-Triangle-s-Club and Edge-ℓ-Triangle-s-Club are defined as follows. Vertex-ℓ-Triangle-s-Club asks to find an s-club S with at least k vertices such that every vertex u ∈ S is part of at least ℓ triangles of G[S]. Similarly, the Edge-ℓ-Triangle-s-Club asks to find an s-club S with at least k vertices such that every edge uv ∈ G[S] is part of at least ℓ triangles of G[S]. In this work, we consider the s-Club problem from the perspective of parameterized complexity as follows. In the first part, we focus on s-Club when parameterized by the size of a given cluster edge deletion set of the graph. We prove that s-Club is FPT with a singly exponential running-time. In the second part, we initiate the study of Vertex-ℓ-Triangle-s-Club and Edge- ℓ-Triangle-s-Club problems with respect to structural parameters. In this work, we provide FPT algorithm for both Vertex-ℓ-Triangle-s-Club and Edge-ℓ-Triangles- Club when parameterized by the size of a vertex cover of the input graph.
