Skeleton matching polytope: Realization and isomorphism

Abstract 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 , S M P ( G ) or, even simpler, S M P . 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 S M P s. Among other simple results, we characterize both planar graphs and bipartite graphs that occur as S M P s. The second question is completely solved here. For every G and H , other than G = G ′ ∪ i = 1 k G i and H = H ′ ∪ i = 1 k H i where, for i = 1 , … , k , G i and H i are isomorphic to K 3 or S 1 , 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.