Removable edges in near-bipartite bricks

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 Lov\'asz 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 K₄ and C₆̄ has at least Δ-2 removable edges. A brick G is near-bipartite if it has a pair of edges ₁,e₂\ such that G-₁,e₂\ 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 |V(G)|-6/2 removable edges. Moreover, all graphs attaining this lower bound are characterized.