Given a rank-r binary matroid we construct a system of $O(r^3)$ linear equations in $O(r^2)$ variables that has a solution over GF(2) if and only if the matroid is graphic.

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preprint (not CWI, to be used with submitted papers)
Matroid Structure for Efficiency
Networks and Optimization

Geelen, J., & Gerards, B. (2011). Characterizing graphic matroids by a system of linear equations. preprint (not CWI, to be used with submitted papers).