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

Abstract

The game of cops and robber is a turn based vertex pursuit game played on a 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 ≥ 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 5 , claw)-free graph is at most 2 that is the conjecture of Sivaraman is true for (P 5 , claw)-free graphs. In this paper, we show that the cop number of a connected (P 5 , H)-free graph is at most 2, where H ∈ {C 4 , C 5 , diamond, paw