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 $e$ such 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 (2007) which is appropriate for the restricted class of near-bipartite bricks. Earlier, the second author 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 second author.

中文翻译：

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生成简单的近二分砖

砖块是一个 $3$-连通图，这样通过删除任何两个不同的顶点从它获得的图具有完美匹配。如果砖块 $G$ 具有一对边 $\alpha$ 和 $\beta$，使得 $G-\{\alpha,\beta\}$ 是二分的并且匹配被覆盖，那么它是接近二分的；例如 $K_4$ 和三棱柱 $\overline{C_6}$。近二分砖的重要性源于匹配覆盖图的耳分解理论。本文的目的是建立一个特定于简单近二分砖类的生成程序。特别地，我们证明了每个简单的近二分块 $G$ 都有一条边 $e$，使得通过收缩与一个二阶顶点相交的每条边从 $Ge$ 获得的图也是一个简单的近二分块砖，除非 $G$ 属于八个明确定义的无限族中的任何一个。这是对 Norine 和 Thomas (2007) 的砖生成定理的改进，适用于近二分砖的受限类。早些时候，第二作者为（不一定是简单的）近二分砖证明了类似的生成定理；我们从这个定理推导出我们的主要结果。我们的证明基于 Carvalho, Lucchesi 和 Murty (2008) 的策略，并使用了他们的一些技术和结果。此处介绍的结果也出现在博士论文中。第二作者的论文。第二作者为（不一定简单）近二分砖证明了类似的生成定理；我们从这个定理推导出我们的主要结果。我们的证明基于 Carvalho, Lucchesi 和 Murty (2008) 的策略，并使用了他们的一些技术和结果。此处介绍的结果也出现在博士论文中。第二作者的论文。第二作者为（不一定简单）近二分砖证明了类似的生成定理；我们从这个定理推导出我们的主要结果。我们的证明基于 Carvalho, Lucchesi 和 Murty (2008) 的策略，并使用了他们的一些技术和结果。此处介绍的结果也出现在博士论文中。第二作者的论文。