Birkhoff--von Neumann Graphs that are PM-Compact

Abstract

A well-studied geometric object in combinatorial optimization is the perfect matching polytope of a graph $G$---the convex hull of the incidence vectors of all perfect matchings of $G$. In any investigation concerning the perfect matching polytope, one may assume that $G$ is matching covered---that is, $G$ is a connected graph (of order at least two) and each edge of $G$ lies in some perfect matching. A graph $G$ is Birkhoff--von Neumann if its perfect matching polytope is characterized solely by nonnegativity and degree constraints. A result of Balas [North Holland Math. Stud., 59 (1981), pp. 1--13] implies that $G$ is Birkhoff--von Neumann 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-$\mathcal{NP}$. However, it is not known to be in $\mathcal{NP}$. The problem is in $\mathcal{P}$ if the input graph is planar---due to a result of Carvalho, Lucchesi, and Murty [J. Combin. Theory Ser. B, 92 (2004), pp. 319--324]. More recently, these authors, along with Kothari [SIAM J. Discrete Math., 32 (2018), pp. 1478--1504], have shown that this problem is equivalent to the seemingly unrelated problem of deciding whether a given graph is $\overline{C_6}$-free. The combinatorial diameter of a polytope is the diameter of its $1$-skeleton graph. A graph $G$ is PM-compact if the combinatorial diameter of its perfect matching polytope equals one. Independent results of Balinski and Russakoff [SIAM Rev., 16 (1974), pp. 516--525] and of Chvátal [J. Combin. Theory Ser. B, 18 (1975), pp. 138--154] imply that $G$ is PM-compact if and only if $G$ does not contain a pair of vertex-disjoint even cycles $(C_1,C_2)$ such that $G-V(C_1)-V(C_2)$ has a perfect matching. Once again the corresponding decision problem is in co-$\mathcal{NP}$, but it is not known to be in $\mathcal{NP}$. The problem is in $\mathcal{P}$ if the input graph is bipartite or is near-bipartite---due to a result of Wang et al. [Discrete Math., 313 (2013), pp. 772--783]. In this paper, we consider the “intersection” of the aforementioned problems. We give an alternative description of matching covered graphs that are Birkhoff--von Neumann as well as PM-compact; our description implies that the corresponding decision problem is in $\mathcal{P}$.