On the P4-structure of perfect graphs III. Partner decompositions

Vertices x and y in the same graph are called partners if there is a set S of three vertices such that both S ∪ {x} and S ∪ {y} induce a chordless path. If the vertices of G can be distributed into two disjoint classes in such a way that every two partners belong to the same class and that each class induces in G a perfect graph then G is perfect. This theorem generalizes two theorems that were proved before, one by Chvátal and Hoàng, and the other by Hoàng alone.