Connected vertex cover for (sP1+ P5) -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 sP2 -free graphs for any integer s≥ 1. We prove that it is also polynomial-time solvable for (sP1+ P5) -free graphs for every integer s≥0.