Subdivided Claws and the Clique-Stable Set Separation Property

Abstract

Let ${\mathscr{C}}$ be a class of graphs closed under taking induced subgraphs. We say that ${\mathscr{C}}$ has the clique-stable set separation property if there exists $c \in {\mathbb{N}}$ such that for every graph $G \in {\mathscr{C}}$ there is a collection ${\mathscr{P}}$ of partitions (X, Y) of the vertex set of G with |${\mathscr{P}}$| ≤ |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 ∩ S = $\emptyset$, then there is (X, Y) ∊ ${\mathscr{P}}$ with K ⊆ X and S ⊆ 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 M. Göös 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 ${\mathscr{S}}, {\mathscr{K}}$of graphs and show that for every S ∊ ${\mathscr{S}}$ and K ∊ ${\mathscr{K}}$, the class of graphs with no induced subgraph isomorphic to S or K has the clique-stable set separation property.