Skeleton matching polytope: Realization and isomorphism

Let G be a simple graph. A set of pairwise disjoint edges is a matching of G. The matching polytope of G, denoted M(G), is the convex hull of the set of the incidence vectors of the matchings of G. The graph G(M(G)), whose vertices and edges are the vertices and edges of M(G), is the skeleton of the matching polytope of G, SMP(G) or, even simpler, SMP. In this paper, we consider the following questions: (1) Which graphs realize SMPs? (2) Given two non-isomorphic graphs, are their skeletons non-isomorphic as well? Concerning the first question, it is well-known (see Naddef and Pulleyblank (1984) and Balinski (1961) [3]) that neither non-connected graphs nor non-Hamiltonian graphs realize SMPs. Among other simple results, we characterize both planar graphs and bipartite graphs that occur as SMPs. The second question is completely solved here. For every G and H, other than G=G′∪i=1kGi and H=H′∪i=1kHi where, for i=1,…,k, Gi and Hi are isomorphic to K3 or S1,3 and G′≃H′, we prove that their skeleton matching polytopes, G(M(G)) and G(M(H)), are isomorphic graphs if and only if G and H are also isomorphic.