  A $3$-connected graph $G$ is a brick if, for any two vertices $u$ and $v$,
the graph $G-\{u,v\}$ has a perfect matching. Deleting an edge $e$ from a brick
$G$ results in a graph with zero, one or two vertices of degree two. The
bicontraction of a vertex of degree two consists of contracting the two edges
incident with it; and the retract of $G-e$ is the graph $J$ obtained from it by
bicontracting all its vertices of degree two. An edge $e$ is thin if $J$ is
also a brick. Carvalho, Lucchesi and Murty [How to build a brick, Discrete
Mathematics 306 (2006), 2383-2410] showed that every brick, distinct from
$K_4$, the triangular prism $\overline{C_6}$ and the Petersen graph, has a thin
edge. Their theorem yields a generation procedure for bricks, using which they
showed that every simple planar solid brick is an odd wheel.
  A brick $G$ is near-bipartite if it has a pair of edges $\alpha$ and $\beta$
such that $G-\{\alpha,\beta\}$ is bipartite and matching covered; examples are
$K_4$ and $\overline{C_6}$. The significance of near-bipartite graphs arises
from the theory of ear decompositions of matching covered graphs.
  The object of this paper is to establish a generation procedure which is
specific to the class of near-bipartite bricks. In particular, we prove that if
$G$ is any near-bipartite brick, distinct from $K_4$ and $\overline{C_6}$, then
$G$ has a thin edge $e$ so that the retract $J$ of $G-e$ is also
near-bipartite.
  In a subsequent work, with Marcelo H. de Carvalho, we use the results of this
paper to prove a generation theorem for simple near-bipartite bricks.
