Excluding Kuratowski graphs and their duals from binary matroids

We consider various applications of our characterization of the internally 4-connected binary matroids with no M(K3,3)-minor. In particular, we characterize the internally 4-connected members of those classes of binary matroids produced by excluding any collection of cycle and bond matroids of K3,3 and K5, as long as that collection contains either M(K3,3) or M ∗(K3,3). We also present polynomial-time algorithms for deciding membership of these classes, where the input consists of a matrix with entries from GF(2). In addition we characterize the maximum-sized simple binary matroids with no M(K3,3)-minor, for any particular rank, and we show that a binary matroid with no M(K3,3)-minor has a critical exponent over GF(2) of at most four.