The perfect matching polytope and solid bricks

The perfect matching polytope of a graph G is the convex hull of the set of incidence vectors of perfect matchings of G. Edmonds (J. Res. Nat. Bur. Standards Sect. B 69B 1965 125) showed that a vector x in Q E belongs to the perfect matching polytope of G if and only if it satisfies the inequalities: (i) x ⩾ 0 (non-negativity), (ii) x ( ∂ ( v ) ) = 1 , for all v ∈ V (degree constraints) and (iii) x ( ∂ ( S ) ) ⩾ 1 , for all odd subsets S of V (odd set constraints). In this paper, we characterize graphs whose perfect matching polytopes are determined by non-negativity and the degree constraints. We also present a proof of a recent theorem of Reed and Wakabayashi.