Near-bipartite bricks in which every b-invariant edge is a forcing edge

A connected graph is matching covered if it has at least one edge and every edge lies in some perfect matching.Lovász proved that every matching covered graph G can be uniquely decomposed into a list of bricks and braces up to multiple edges.

Denote by b(G) the number of bricks in such a decomposition.

An edge e of G is removable if G-e is also matching covered; is b-invariant if e is removable and b(G-e)=b(G).

Furthermore, an edge e of G is a forcing edge if it lies in precisely one perfect matching of G.

Lucchesi and Murty proposed the problem of characterizing bricks, distinct from K_4, \overline{C_6}, and the Petersen graph, in which every b-invariant edge is a forcing edge.

In this paper, we solve this problem for near-bipartite bricks by providing a complete characterization.