A graph G is said to be 2-divisible if for each induced subgraph H of G, either V(H) is a stable set or V(H) can be partitioned into two sets A and B such that \(\omega (H[A])< \omega (H)\) and \(\omega (H[B])< \omega (H)\) . A graph G is called perfectly divisible if for each induced subgraph H of G, the set V(H) can be partitioned into two sets A and B such that H[A] is perfect and \(\omega (H[B])< \omega (H)\) . A graph \(P_{2}\cup P_{3}\) is the disjoint union of paths \(P_{2}\) and \(P_{3}\) . A bull is a graph consisting of a triangle with two disjoint pendant edges. In this paper, we prove that (a) each \((C_{5}, P_{2}\cup P_{3})\) -free graph is 2-divisible;
(b) a \((bull, P_{2}\cup P_{3})\) -free graph G with \(\omega (G)\ge 3\) has a partition (X, Y) such that G[X] is perfect and \(\omega (G[Y])< \omega (G)\) if G admits no homogeneous set.
