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

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