The connected vertex cover problem in k-regular graphs

Abstract

Given a connected graph \(G=(V,E)\), the Connected Vertex Cover (CVC) problem is to find a vertex set \(S\subset V\) with minimum cardinality such that every edge is incident to a vertex in S, and moreover, the induced graph G[S] is connected. In this paper, we investigate the CVC problem in k-regular graphs for any fixed k (\(k\ge 4\)). First, we prove that the CVC problem is NP-hard for k-regular graphs,and then we give a lower bound for the minimum size of a CVC, based on which, we propose a \(\frac{2k}{k+2}+O(\frac{1}{n})\)-approximation algorithm for the CVC problem.