Given a rank-$r$ binary matroid we construct a system of $O(r^3)$ linear equations in $O(r^3)$ variables that has a solution over GF$(2)$ if and only if the matroid is graphic.
Matroids, Graphc Matroids, Planar Graphs
Matroids, geometric lattices (msc 05B35)
Other (theme 6)
Academic Press
doi.org/10.1016/j.jctb.2013.07.00
Journal of Combinatorial Theory - Series B
Networks and Optimization

Geelen, J, & Gerards, A.M.H. (2013). Characterizing graphic matroids by a system of linear equations. Journal of Combinatorial Theory - Series B, 103(5), 642–646. doi:10.1016/j.jctb.2013.07.00