Theorem proves that M has a 3-connected proper minor N with |E(M) − E(N) | = 1 unless M is a wheel or a whirl. This paper establishes a corresponding result for internally 4-connected binary matroids. In particular, we prove that if M is such a matroid, then M has an internally 4-connected proper minor N with |E(M) − E(N) | ≤ 3 unless M or its dual is the cycle matroid of a planar or Möbius quartic ladder, or a 16-element variant of such a planar ladder. 1.

Abstract. Let M be a 3-connected matroid other than a wheel or a whirl. In the next paper in this series, we prove that there is an element whose deletion from M or M ∗ is 3-connected and whose only 3-separations are equivalent to those induced by M. The strategy used to prove this theorem involves showing that we can remove some element from a leaf of the tree of 3-separations of M. The main result of this paper is designed to allow us to do this. 1.