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. We provide FPT algorithm for both Vertex-l-Triangle-s-Club and Edge-l-Triangle-s-Club when parameterized by the size of a vertex cover of the input graph. In the second part, we focus on Edge-l-Triangle-2-Club and Vertex-l-Triangle-2- Club parameterized by the size of the cluster vertex deletion set. In the third part, we provide a FPT algorithm for Vertex-l-Triangle-2-Club parameterized by the dual parameters treewidth and number of triangle. We provide a FPT algorithm and a compression kernel for s-Club parameterized by cluster edge deletion and provide a linear kernel for Vertex-l-Triangle-s-Club parameterized by feedback edge set.
