Generating bricks

13 years 4 months ago

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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 Lov´asz and Plummer. We prove a “splitter theorem” for bricks. More precisely, we show that if a brick H is a “matching minor” of a brick G, then, except for a few well-described exceptions, a graph isomorphic to H can be obtained from G by repeatedly applying a certain operation in such a way that all the intermediate graphs are bricks and have no parallel edges. The operation is as follows: ﬁrst delete an edge, and for every vertex of degree two that results contract both edges incident with it. This strengthens a recent result of de Carvalho, Lucchesi and Murty. 8 March 2003, revised 5 December 2006. 1 Partially supported by NSF under Grant No. DMS-0200595. 2 Partially supported by NSF under Grant No. DMS-0200595 an...

Serguei Norine, Robin Thomas

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