  In a graph, a Clique-Stable Set separator (CS-separator) is a family
$\mathcal{C}$ of cuts (bipartitions of the vertex set) such that for every
clique $K$ and every stable set $S$ with $K \cap S = \emptyset$, there exists a
cut $( W,W')$ in $\mathcal{C}$ such that $K \subseteq W$ and $S \subseteq W'$.
Starting from a question concerning extended formulations of the Stable Set
polytope and a related complexity communication problem, Yannakakis [17] 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.
