Removable edges in near-bipartite bricks

An edge e $e$ of a matching covered graph G $G$ is removable if G - e $G-e$ is also matching covered. The notion of removable edge arises in connection with ear decompositions of matching covered graphs introduced by Lov & aacute;sz and Plummer. A nonbipartite matching covered graph G $G$ is a brick if it is free of nontrivial tight cuts. Carvalho, Lucchesi and Murty proved that every brick other than K 4 ${K}_{4}$ and C 6 <overline> $\bar{{C}_{6}}$ has at least Delta - 2 ${\rm{\Delta }}-2$ removable edges. A brick G $G$ is near-bipartite if it has a pair of edges {e 1 , e 2 } $\{{e}_{1},{e}_{2}\}$ such that G -{e 1 , e 2 } $G-\{{e}_{1},{e}_{2}\}$ is a bipartite matching covered graph. In this paper, we show that in a near-bipartite brick G $G$ with at least six vertices, every vertex of G $G$, except at most six vertices of degree three contained in two disjoint triangles, is incident with at most two nonremovable edges; consequently, G $G$ has at least | V( G ) | - 6 2 $\frac{|V(G)|-6}{2}$ removable edges. Moreover, all graphs attaining this lower bound are characterized.