Decomposition techniques applied to the Clique-Stable set separation problem

In a graph, a Clique-Stable Set separator (CS-separator) is a family C of cuts (bipartitions of the vertex set) such that for every clique K and every stable set S with K∩S=∅, there exists a cut (W,W′) in C such that K⊆W and S⊆W′. Starting from a question concerning extended formulations of the Stable Set polytope and a related complexity communication problem, Yannakakis (Yannakakis, 1991) asked in 1991 the following questions: does every graph admit a polynomial-size CS-separator? If not, does every perfect graph do? Several positive and negative results related to this question were given recently. Here we show how graph decomposition can be used to prove that a class of graphs admits a polynomial CS-separator. We apply this method to apple-free graphs and cap-free graphs.