The perfectly matchable subgraph polytope of a bipartite graph

Abstract

Abstract The following type of problem arises in practice: in a node‐weighted graph G , find a minimum‐weight node set that satisfies certain conditions and, in addition, induces a perfectly matchable subgraph of G. This has led us to study the convex hull of incidence vectors of node sets that induce perfectly matchable subgraphs of a graph G , which we call the perfectly matchable subgraph polytope of G. For the case when G is bipartite, we give a linear characterization of this polytope, i.e., specify a system of linear inequalities whose basic solutions are the incidence vectors of perfectly matchable node sets of G. We derive this result by three different approaches, using linear programming duality, projection, and lattice polyhedra, respectively. The projection approach is used here for the first time as a proof method in polyhedral combinatorics, and seems to have many similar applications. Finally, we completely characterize the facets of our polytope; i.e., we separate the essential inequalities of our linear defining system from the redundant ones.