Matching polytopes, Gorensteinness, and the integer decomposition property

The matching polytope of a graph G is the convex hull of the indicator vectors of the matchings on G. We characterize the graphs whose associated matching polytopes are Gorenstein, and then prove that all Gorenstein matching polytopes possess the integer decomposition property. As a special case study, we examine the matching polytopes of wheel graphs and show that they are not Gorenstein, but do possess the integer decomposition property.