A chain theorem for internally 4-connected binary matroids

Abstract

Let M be a matroid. When M is 3-connected, Tutte\u27s Wheels-and-Whirls 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. © 2011 Elsevier Inc