Birkhoff-von Neumann Graphs that are PM-compact
Marcelo H. de Carvalho, Nishad Kothari, Xiumei Wang, Yixun Lin

TL;DR
This paper characterizes graphs that are both Birkhoff-von Neumann and PM-compact, providing a polynomial-time decision procedure for these properties by exploring their combinatorial and polyhedral structures.
Contribution
It offers a complete characterization of graphs that are both BvN and PM-compact, enabling efficient decision algorithms for these properties.
Findings
Characterization of graphs that are both BvN and PM-compact.
Decision problem for these properties is in P.
Connection between graph properties and polyhedral diameters.
Abstract
A well-studied geometric object in combinatorial optimization is the perfect matching polytope of a graph . In any investigation concerning the perfect matching polytope, one may assume that 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 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 is BvN if and only if does not contain a pair of vertex-disjoint odd cycles such that 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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Computational Geometry and Mesh Generation · Graph Labeling and Dimension Problems
