Cops and robber on subclasses of $P_5$-free graphs

TL;DR

This paper investigates the minimum number of cops needed to catch a robber on certain classes of graphs, proving that for ($P_5$, H)-free graphs with specific H, only two cops are sufficient.

Contribution

It establishes that the cop number is at most 2 for a broad class of ($P_5$, H)-free graphs, extending previous bounds and confirming conjectures for these subclasses.

Findings

Cop number of ($P_5$, H)-free graphs is at most 2 for specified H.

Extends bounds on cop number for subclasses of $P_5$-free graphs.

Supports Sivaraman's conjecture within these subclasses.

Abstract

The game of cops and robber is a turn based vertex pursuit game played on a connected graph between a team of cops and a single robber. The cops and the robber move alternately along the edges of the graph. We say the team of cops win the game if a cop and the robber are at the same vertex of the graph. The minimum number of cops required to win in each component of a graph is called the cop number of the graph. Sivaraman [Discrete Math. 342(2019), pp. 2306-2307] conjectured that for every $t\geq 5$, the cop number of a connected $P_t$-free graph is at most $t-3$, where $P_t$ denotes a path on $t$~vertices. Turcotte [Discrete Math. 345 (2022), pp. 112660] showed that the cop number of any $2K_2$-free graph is at most $2$, which was earlier conjectured by Sivaraman and Testa. Note that if a connected graph is $2K_2$-free, then it is also $P_5$-free. Liu showed that the cop number of a connected ($P_t$, $H$)-free graph is at most $t-3$, where $H$ is a cycle of length at most $t$ or a claw. So the conjecture of Sivaraman is true for ($P_5$, $H$)-free graphs, where $H$ is a cycle of length at most $5$ or a claw. In this paper, we show that the cop number of a connected ($P_5,H$)-free graph is at most $2$, where $H\in \{C_4$, $C_5$, diamond, paw, $K_4$, $2K_1\cup K_2$, $K_3\cup K_1$, $P_3\cup P_1\}$.