  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.
