Toric rings and ideals of stable set polytopes

TL;DR

This paper investigates the algebraic properties of toric rings and ideals associated with stable set polytopes, providing characterizations and criteria for normality and generators, especially for graphs with stability number two.

Contribution

It offers a graph-theoretical characterization of generators and a normality criterion for toric rings of stable set polytopes, extending results from edge polytopes.

Findings

Characterization of generators for graphs with stability number two

Criterion for normality of toric rings of stable set polytopes

Identification of infinite families with quadratic generators but no quadratic Gr"obner bases

Abstract

In this paper, we discuss the normality of the toric rings of stable set polytopes, and the set of generators and Gr\"obner bases of toric ideals of stable set polytopes by using the results on that of edge polytopes of finite nonsimple graphs. In particular, for a graph of stability number two, we give a graph theoretical characterization of the set of generators of the toric ideal of the stable set polytope, and a criterion to check whether the toric ring of the stable set polytope is normal or not. One of the application of the results is an infinite family of stable set polytopes whose toric ideal is generated by quadratic binomials and has no quadratic Gr\"obner bases.