Cop number of $P_k$-free graphs without a claw or a fork

Abstract

In 2019, Liu proved that the cop number of any $(P_k, claw)$-free graph is at most $k-3$. In 2026, Clow and Meger improved and generalized the result of Liu by showing that every connected $(P_k, E)$-free graph has cop number at most $\lceil \frac{k-1}{2} \rceil + 3$, where the graph $E$ is obtained by subdividing two edges of a claw. In this paper, we show that every $(P_k, claw)$-free graph $G$ has cop number at most $\lceil \frac{k-1}{2} \rceil$. This improves the result of Clow and Meger on claw-free graphs. We further show that this bound can be improved to $\lceil \frac{k}{3}\rceil$, when $G$ contains a hole of length at least $k-1$. We show that the cop number of any $(P_k, fork)$-free graph is at most $\lceil \frac{2(k-1)}{3} \rceil$. This also improves the result of Clow and Meger on fork-free graphs for some values of $k$. We further show that every $(P_7, fork)$-free graph has cop number at most $3$, and this bound is tight.