Vector Spaces and the Petersen Graph - de Carvalho, Marcelo H.; Little, C. H. C. (The Electronic Journal of Combinatorics, 2008)
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 $|S \cap A|$ is even for every alternating circuit $A$ but $|S \cap C|$ 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.