  A graph $G$ is said to be $2$-divisible if for all (nonempty) induced
subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A,B$ such that
$\omega(A) < \omega(H)$ and $\omega(B) < \omega(H)$. A graph $G$ is said to be
perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be
partitioned into two sets $A,B$ such that $H[A]$ is perfect and $\omega(B) <
\omega(H)$. We prove that if a graph is $(P_5,C_5)$-free, then it is
$2$-divisible. We also prove that if a graph is bull-free and either
odd-hole-free or $P_5$-free, then it is perfectly divisible.
