Cops and robbers on $P_5$-free graphs

TL;DR

This paper proves that all connected $P_5$-free graphs can be captured with at most two cops, confirming a conjecture and revealing structural properties of such graphs related to their independence number.

Contribution

It establishes that the cop number of connected $P_5$-free graphs is at most two and characterizes their structure when the independence number is at least three.

Findings

Connected $P_5$-free graphs have cop number at most two.

Structural characterization of $P_5$-free graphs with independence number ≥ 3.

Confirmed a conjecture of Sivaraman.

Abstract

We prove that every connected $P_5$-free graph has cop number at most two, solving a conjecture of Sivaraman. In order to do so, we first prove that every connected $P_5$-free graph $G$ with independence number at least three contains a three-vertex induced path with vertices $a \hbox{-} b \hbox{-} c$ in order, such that every neighbour of $c$ is also adjacent to one of $a,b$.