Removable edges in near‐bipartite bricks

Abstract

An edge of a matching covered graph is removable if is also matching covered. The notion of removable edge arises in connection with ear decompositions of matching covered graphs introduced by Lovász and Plummer. A nonbipartite matching covered graph is a brick if it is free of nontrivial tight cuts. Carvalho, Lucchesi and Murty proved that every brick other than and has at least removable edges. A brick is near‐bipartite if it has a pair of edges such that is a bipartite matching covered graph. In this paper, we show that in a near‐bipartite brick with at least six vertices, every vertex of , except at most six vertices of degree three contained in two disjoint triangles, is incident with at most two nonremovable edges; consequently, has at least removable edges. Moreover, all graphs attaining this lower bound are characterized.