Generating simple near-bipartite bricks

A brick is a $3$-connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. 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 the triangular prism $\overline{C_6}$. The significance of near-bipartite bricks 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 simple near-bipartite bricks. In particular, we prove that every simple near-bipartite brick $G$ has an edge $e$ so that the graph obtained from $G-e$ by contracting each edge that is incident with a vertex of degree two is also a simple near-bipartite brick, unless $G$ belongs to any of eight well-defined infinite families. This is a refinement of the brick generation theorem of Norine and Thomas (Generating Bricks, J. Combin. Theory Ser. B, 2007) which is appropriate for the restricted class of near-bipartite bricks. Earlier, the first author (Generating near-bipartite bricks, J. Graph Theory, 2019) proved a similar generation theorem for (not necessarily simple) near-bipartite bricks; we deduce our main result from this theorem. Our proof is based on the strategy of Carvalho, Lucchesi and Murty (2008) and uses several of their techniques and results. The results presented here also appear in the Ph.D. thesis of the first author.