Excluding Kuratowski graphs and their duals from binary matroids

We consider some applications of our characterisation of the internally 4-connected binary matroids with no M(K3,3)-minor. We characterise the internally 4-connected binary matroids with no minor in M, where M is a subset of {M(K3,3),M⁎(K3,3),M(K5),M⁎(K5)} that contains either M(K3,3) or M⁎(K3,3). We also describe a practical algorithm for testing whether a binary matroid has a minor in M. In addition we characterise the growth-rate of binary matroids with no M(K3,3)-minor, and we show that a binary matroid with no M(K3,3)-minor has critical exponent over GF(2) at most equal to four.