A class of non-binary matroids with many binary minors Original Research Article

It is a well-known result of Tutte, A homotopy theorem for matroids, I, II, Trans. Amer. Math. Soc. 88 (1958) 144–174. that U2,4 is the only non-binary matroid M such that, for every element e, both M⧹e and M/e are binary. Oxley generalized this result by characterizing the non-binary matroids M such that, for every element e of M, the deletion M⧹e or the contraction M/e is binary. We characterize those non-binary matroids M such that, for all elements e and f, at least two of M⧹e,f; M⧹e/f; M/e⧹f; and M/e,f are binary.