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 tβ₯5, the cop number of a connected
Ptβ-free graph is at most tβ3, where Ptβ denotes a path on t~vertices.
Turcotte [Discrete Math. 345 (2022), pp. 112660] showed that the cop number of
any 2K2β-free graph is at most 2, which was earlier conjectured by
Sivaraman and Testa. Note that if a connected graph is 2K2β-free, then it is
also P5β-free. Liu showed that the cop number of a connected (Ptβ,
H)-free graph is at most tβ3, where H is a cycle of length at most t or
a claw. So the conjecture of Sivaraman is true for (P5β, 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 (P5β,H)-free graph is at most 2, where
Hβ{C4β, C5β, diamond, paw, K4β, 2K1ββͺK2β, K3ββͺK1β,
P3ββͺP1β}