The stable set polytope of ($P_6$,triangle)-free graphs and new facet-inducing graphs

TL;DR

This paper provides a complete characterization of the stable set polytope for ($P_6$,triangle)-free graphs, introducing new facet-inducing graphs and leveraging structural and computational tools to advance understanding in combinatorial optimization.

Contribution

It offers a full description of the stable set polytope for specific graph classes and identifies new facet-inducing graphs using structural analysis and software tools.

Findings

Complete description of STAB($G$) for ($P_6$,triangle)-free graphs

Identification of new facet-inducing graphs

Structural results on prime ($P_6$,triangle)-free graphs

Abstract

The stable set polytope of a graph $G$, denoted as STAB($G$), is the convex hull of all the incidence vectors of stable sets of $G$. To describe a linear system which defines STAB($G$) seems to be a difficult task in the general case. In this paper we present a complete description of the stable set polytope of ($P_6$,triangle)-free graphs (and more generally of ($P_6$,paw)-free graphs). For that we combine different tools, in the context of a well known result of Chv\'atal \cite{Chvatal1975} which allows to focus just on prime facet-inducing graphs, with particular reference to a structure result on prime ($P_6$,triangle)-free graphs due to Brandst\"adt et al. \cite{BraKleMah2005}. Also we point out some peculiarities of new facet-inducing graphs detected along this study with the help of a software.