Generating simple near‐bipartite bricks

Abstract

AbstractA 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 is near‐bipartite if it has a pair of edges and such that is bipartite and matching covered; examples are and the triangular prism . 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 has an edge such that the graph obtained from by contracting each edge that is incident with a vertex of degree two is also a simple near‐bipartite brick, unless belongs to any of eight well‐defined infinite families. This is a refinement of the brick generation theorem of Norine and Thomas which is applicable to the class of near‐bipartite bricks. Earlier, the first 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 a strategy of Carvalho, Lucchesi and Murty and uses several of their techniques and results.