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.
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References
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Chudnovsky, M., Seymour, P. (2021). Subdivided Claws and the Clique-Stable Set Separation Property. In: de Gier, J., Praeger, C.E., Tao, T. (eds) 2019-20 MATRIX Annals. MATRIX Book Series, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-030-62497-2_29
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DOI: https://doi.org/10.1007/978-3-030-62497-2_29
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