Cops and robber on subclasses of P_5-free graphs
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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, K_4, 2K_1∪ K_2, K_3∪ K_1, P_3∪ P_1}.
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