Connected Vertex Cover for $$(sP_1+P_5)$$-Free Graphs

Abstract

The Connected Vertex Cover problem is to decide if a graph G has a vertex cover of size at most k that induces a connected subgraph of G. This is a well-studied problem, known to be NP-complete for restricted graph classes, and, in particular, for H-free graphs if H is not a linear forest. On the other hand, the problem is known to be polynomial-time solvable for \(sP_2\)-free graphs for any integer \(s\ge 1\). We prove that it is also polynomial-time solvable for \((sP_1+P_5)\)-free graphs for every integer \(s\ge ~0\).