Subdivided Claws and the Clique-Stable Set Separation Property

TL;DR

This paper investigates the clique-stable set separation property in graph classes, defining specific families of graphs and proving that classes excluding certain induced subgraphs possess this property.

Contribution

It introduces two infinite graph families and proves classes excluding these subgraphs have the clique-stable set separation property.

Findings

Identifies two infinite families of graphs relevant to the property.

Shows classes excluding these graphs have the clique-stable set separation property.

Abstract

Let $\mathcal{C}$ be a class of graphs closed under taking induced subgraphs. We say that $\mathcal{C}$ has the {\em clique-stable set separation property} if there exists $c \in \mathbb{N}$ such that for every graph $G \in \mathcal{C}$ there is a collection $\mathcal{P}$ of partitions $(X,Y)$ of the vertex set of $G$ with $|\mathcal{P}| \leq |V(G)|^c$ and with the following property: if $K$ is a clique of $G$, and $S$ is a stable set of $G$, and $K \cap S =\emptyset$, then there is $(X,Y) \in \mathcal{P}$ with $K \subseteq X$ and $S \subseteq Y$. In 1991 M. Yannakakis conjectured that the class of all graphs has the clique-stable set separation property, but this conjecture was disproved by G\"{o}\"{o}s in 2014. Therefore it is now of interest to understand for which classes of graphs such a constant $c$ exists. In this paper we define two infinite families $\mathcal{S}, \mathcal{K}$ of graphs and show that for every $S \in \mathcal{S}$ and $K \in \mathcal{K}$, the class of graphs with no induced subgraph isomorphic to $S$ or $K$ has the clique-stable set separation property.