Brick Generation

A brick is a 3-connected graph such that the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of the matching decomposition procedure of Kotzig, and Lovász and Plummer. We prove that every brick except for the Petersen graph can be reduced to K4 or the prism by repeatedly applying a certain reduction operation in such a way that all the intermediate graphs are bricks. The reduction operation is as follows: first delete an edge, and for every vertex of degree two that results contract both edges incident with it and delete all parallel edges thus produced. In fact, our main theorem is stronger in two respects. First, rather than starting with K4 or the prism, the generation process for a brick G can start with any (suitably defined) “minor” of G. Second, except for a few well-described exceptional graphs, the reduction operation can be carried out in such a way that no parallel edges are created. 8 March 2003, revised 1 March 2004. This is a preliminary version. Please check http://www.math.gatech.edu/ ̃thomas/ for the latest revision. 1 Partially supported by NSF under Grant No. DMS-0200595. 2 Partially supported by NSF under Grant No. DMS-0200595 and by ONR under Grant No. N00014-