Vector spaces and the Petersen graph

Abstract

It is shown that a matching covered graph has an ear decomposition with no more than one double ear if and only if there is no set S of edges such that |S∩A| is even for every alternating circuit A but ∪S∩ C| is odd for some even circuit C. Two proofs are presented. The first uses vector spaces and the second is constructive. Some applications are also given.