Stable set polytopes and their 1-skeleta

TL;DR

This paper characterizes the edges of certain 0/1-polytopes, including stable set polytopes and related classes, and explores their structure, hierarchy, and diameter, providing insights into their combinatorial properties.

Contribution

It introduces a new characterization of edges for specific 0/1-polytopes and analyzes their position within the hierarchy of such polytopes, extending understanding of their combinatorial structure.

Findings

Characterization of edges for stable set polytopes and related classes

Analysis of the hierarchy of 0/1-polytopes including matroid polytopes

Improved bounds on the diameter of these polytopes

Abstract

We characterize the edges of two classes of $0/1$-polytopes. The first class corresponds to the stable set polytope of a graph $G$ and includes chain polytopes of posets, some instances of matroid independence polytopes, as well as newly-defined polytopes whose vertices correspond to noncrossing set partitions. In analogy with matroid basis polytopes, the second class is obtained by considering the stable sets of maximal cardinality. We investigate how the class of $0/1$-polytopes whose edges satisfy our characterization is situated within the hierarchy of $0/1$-polytopes. This includes the class of matroid polytopes. We also study the diameter of these classes of polytopes and improve slightly on the Hirsch bound.