Cops and robbers on $P_5$-free graphs
Maria Chudnovsky, Sergey Norin, Paul Seymour, Jérémie Turcotte
TL;DR
The paper resolves Sivaraman's conjecture by proving that every connected $P_5$-free graph satisfies $c(G) \le 2$. The authors develop a structural framework centered on domineering 3-paths, bijoined graphs, and $P_3$-connected subgraphs, culminating in a two-cop strategy that necessarily captures the robber. A key step is proving that any connected $P_5$-free graph with $\alpha(G) \ge 3$ contains a domineering $a$-$b$-$c$ path with $N[c]\subseteq N[a]\cup N[b]$, and ruling out potential obstructions via a bijoined-graph analysis. The result advances understanding of pursuit-evasion on forbidden-subgraph classes and settles a central open question in the area.
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 $a,b$.