Integer decomposition property for Cayley sums of order and stable set polytopes

TL;DR

This paper investigates the integer decomposition property (IDP) of Cayley sums involving order and stable set polytopes, establishing conditions under which these sums are Gorenstein and possess IDP, especially for perfect graphs.

Contribution

It proves that the Cayley sum of an order polytope and a stable set polytope of a perfect graph has IDP and a unimodular triangulation, linking Gorenstein properties to graph perfection.

Findings

Cayley sum of an order polytope and a stable set polytope of a perfect graph has IDP.

Cayley sum and Minkowski sum are Gorenstein if and only if the graph is perfect.

The Cayley sum of these polytopes admits a regular unimodular triangulation.

Abstract

Lattice polytopes which possess the integer decomposition property (IDP for short) turn up in many fields of mathematics. It is known that if the Cayley sum of lattice polytopes possesses IDP, then so does their Minkowski sum. In this paper, the Cayley sum of the order polytope of a finite poset and the stable set polytope of a finite simple graph is studied. We show that the Cayley sum of an order polytope and the stable set polytope of a perfect graph possesses a regular unimodular triangulation and IDP, and hence so does their Minkowski sum. Moreover, it turns out that, for an order polytope and the stable set polytope of a graph, the following conditions are equivalent: (i) the Cayley sum is Gorenstein; (ii) the Minkowski sum is Gorenstein; (iii) the graph is perfect.