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

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 Möbius 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 M \ x 3 has an N -minor and is ( 4 , 4 , S ) -connected. We also showed that, when M has a good bowtie, either M \ x 3 , x 6 has an N -minor and M \ x 6 is ( 4 , 4 , S ) -connected; or M \ x 3 / x 2 has an N -minor and is ( 4 , 4 , S ) -connected. In this paper, we show that, when M \ x 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.