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

TL;DR

This paper proves that the Connected Vertex Cover problem can be solved in polynomial time for a new class of graphs called $(sP_1+P_5)$-free graphs, extending known results for simpler graph classes.

Contribution

It establishes polynomial-time solvability of the Connected Vertex Cover problem for $(sP_1+P_5)$-free graphs, broadening the understanding of tractable graph classes.

Findings

Polynomial-time algorithm for $(sP_1+P_5)$-free graphs

Extension of tractability from $P_4$-free graphs

Advances understanding of graph classes with polynomial solutions

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 (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 $P_4$-free graphs. We continue the search for tractable graph classes: we prove that it is also polynomial-time solvable for $(sP_1+P_5)$-free graphs for every integer $s\geq 0$.