Removable edges in near-bipartite bricks

Abstract

An edge $e$ of a matching covered graph $G$ is removable if $G-e$ 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 $G$ is a brick if it is free of nontrivial tight cuts. Carvalho, Lucchesi and Murty proved that every brick other than $K_4$ and $\overline{C_6}$ has at least $\Delta -2$ removable edges. A brick $G$ is near-bipartite if it has a pair of edges $\left\{e_1,e_2\right\}$ such that $G-\left\{e_1,e_2\right\}$ is a bipartite matching covered graph. In this paper, we show that in a near-bipartite brick $G$ with at least six vertices, every vertex of $G$, except at most six vertices of degree three contained in two disjoint triangles, is incident with at most two nonremovable edges; consequently, $G$ has at least $\frac{\left|V\left(G\right)\right|-6}{2}$ removable edges. Moreover, all graphs attaining this lower bound are characterized.