Cop number of $2K_2$-free graphs

We prove that the cop number of a $2K_2$-free graph is at most $2$ if it has diameter $3$ or does not have an induced cycle of length $k$, where $k \ \in {3,4,5}$\mathrm{.}Weconjecturethatthecopnumberofevery$2K_2$-free graph is at most $2$.