Towards a splitter theorem for internally 4-connected binary matroids VII

摘要

Let M be a 3-connected binary matroid; M is internally 4-connected if one side of every 3-separation is a triangle or a triad, and M is (4, 4, S)-connected if one side of every 3-separation is a triangle, a triad, or a 4-element fan. Assume M is internally 4-connected and that neither M nor its dual is a cubic Mobius or planar ladder or a certain coextension thereof. Let N be an internally 4-connected proper minor of M. Our aim is to show that M has a proper internally 4-connected minor with an N-minor that can be obtained from M either by removing at most four elements, or by removing elements in an easily described way from a special substructure of M. When this aim cannot be met, the earlier papers in this series showed that, up to duality, M has a good bowtie, that is, a pair, {x(1), x(2), x(3)} and {x(4), x(5), x(6)}, of disjoint triangles and a cocircuit, {x(2), x(3), x(4), x(5)}, where Mx(3) has an N-minor and is (4,4, S)-connected. We also showed that, when M has a good bowtie, either Mx(3), x(6) has an N-minor and Mx(6) is (4,4, S)-connected; or Mx(3)/x(2) has an N-minor and is (4, 4, S)-connected. In this paper, we show that, when Mx(3), x(6) has no N-minor, M has an internally 4-connected proper minor with an N-minor that can be obtained from M by removing at most three elements, or by removing elements in a well-described way from a special substructure of M. This is a final step towards obtaining a splitter theorem for the class of internally 4-connected binary matroids. (C) 2018 Elsevier Inc. All rights reserved.