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 give a polynomial-time algorithm to solve the problem for $$(sP_1+P_5)$$-free graphs for every integer $$s\ge ~0$$. Our algorithm can also be used for the Weighted Connected Vertex Cover problem.