The key to Seymour's Regular Matroid Decomposition Theorem is his result that each 3-connected regular matroid with no $R_{10}$- or $R_{12}$-minor is graphic or cographic. We present a proof of this in terms of signed graphs

, , ,
Wiley
Journal of Graph Theory
Probability, Networks and Algorithms

Geelen, J., & Gerards, B. (2005). Regular matroid decomposition via signed graphs. Journal of Graph Theory, 48, 74–84.