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 (a graph is H-free if it does not contain H as an induced subgraph). It is easy to see that Connected Vertex Cover is polynomial-time solvable for P4-free graphs. We continue the search for tractable graph classes: we prove that it is also polynomial-time solvable for (sP1+P5)-free graphs for every integer s≥0.
@article{arxiv.1712.08362,
title = {Connected Vertex Cover for $(sP_1+P_5)$-Free Graphs},
author = {Matthew Johnson and Giacomo Paesani and Daniel Paulusma},
journal= {arXiv preprint arXiv:1712.08362},
year = {2018}
}