A geometric object of great interest in combinatorial optimization is the perfect matching polytope of a graph $G$. In any investigation concerning the perfect matching polytope, one may assume that $G$ is matching covered --- that is, it is a connected graph (of order at least two) and each edge lies in some perfect matching. A graph $G$ is Birkhoff-von Neumann (BvN) if its perfect matching polytope is characterized solely by non-negativity and degree constraints. A result of Balas (1981) implies that $G$ is BvN if and only if $G$ does not contain a pair of vertex-disjoint odd cycles $(C_1,C_2)$ such that $G-V(C_1)-V(C_2)$ has a perfect matching. It follows immediately that the corresponding decision problem is in co-NP. However, it is not known to be in NP. The problem is in P if the input graph is planar --- due to a result of Carvalho, Lucchesi and Murty (2004). These authors, along with Kothari (2018), have shown that this problem is equivalent to the seemingly unrelated problem o 查看全文>>