# Decomposition techniques applied to the Clique-Stable set Separation problem

## TL;DR

This paper investigates the use of graph decomposition techniques to establish the existence of polynomial-size Clique-Stable set separators in specific graph classes, advancing understanding of their structural properties.

## Contribution

It introduces a novel method applying graph decomposition to prove polynomial CS-separators for apple-free and cap-free graphs.

## Key findings

- Polynomial CS-separators exist for apple-free graphs.
- Polynomial CS-separators exist for cap-free graphs.
- Graph decomposition is effective for analyzing CS-separators.

## Abstract

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.