Clique-Stable Set separation in perfect graphs with no balanced skew-partitions

We prove that perfect graphs with no balanced skew-partitions admits a O(n) Clique-Stable set separator. We then prove that for every graph G in this class, there always exists two subsets of vertices V1, V2 such that |V1|, |V2| ≥ c|V (G)| for some constant c and V1 is either complete or anticomplete to V2. The method uses trigraphs, introduced by Chudnovsky [5], and relies on the good behavior of both problems with respect to 2-join decomposition proved by Chudnovsky, Trotignon, Trunck and Vušković [7]. Moreover we define the generalized k-join and generalize both our results on classes of graphs admitting such a decomposition.