A Characterization of Weakly Bipartite Graphs

A labeled graph is said to be weakly bipartite if the clutter of its odd cycles is ideal. Seymour conjectured that a labeled graph is weakly bipartite if and only if it does not contain a minor called an oddK5. A proof of this conjecture is given in this paper. Let G = (V;E) be a graph and E. We call edges in odd and edges in E even. The pair (G; ) is called a labeled graph. A subset L E(G) is odd (resp. even) if jL\ j is odd (resp. even). A cycle of G is a connected subgraph of G with all degrees equal to two. We say that a labeled graph (G; ) is weakly bipartite if the following polyhedron Q is integral (i.e. all its extreme points are integral): Q = x 2 <jEj + :Xi2C xi 1; for all odd cycles C of (G; ) (1) The case where = E(G) is of particular interest. Let x̂ be any 0; 1 extreme point of Q. Then x̂ is the incidence vector of a set of edges which intersects every odd cycle of G. In other words e x̂ is the incidence vector of a bipartite subgraph of G. Let w 2 <jEj + be weights for the edges of G and let x be a solution to min wx : x 2 Q \ f0; 1gjEj : (2) Then e x is a solution to the Weighted Max-Cut problem. This problem is known to be NP-Hard even in the unweighted case [6]. Notice that weakly bipartite graphs are precisely those graphs for which the integrality constraints in (2) can be dropped. Weakly bipartite graphs G with = E(G) where introduced by Grötschel and Pulleyblank [5] which showed that the optimization problem minfwx : x 2 Qg can be solved in polynomial time and which proved that planar graphs are weakly bipartite. Date: November 97.