On the structure of perfectly divisible graphs

A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ contains a set $X$ of vertices such that $X$ meets all largest cliques of $H$, and $X$ induces a perfect graph. The chromatic number of a perfectly divisible graph $G$ is bounded by $\omega^2$ where $\omega$ denotes the number of vertices in a largest clique of $G$. A graph $G$ is minimally non-perfectly divisible if $G$ is not perfectly divisible but each of its proper induced subgraph is. A set $C$ of vertices of $G$ is a clique cutset if $C$ induces a clique in $G$, and $G-C$ is disconnected. We prove that a $P_5$-free minimally non-perfectly divisible graph cannot contain a clique cutset. This result allows us to re-establish several theorems on the perfect divisibility of some classes of $P_5$-free graphs. We will show that recognizing perfectly divisible graphs is NP-hard.