2016/08/02 by Chun, Carolyn, Oxley, James
#05B35 #05C40 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1608.01027
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, \x1,x2,x3\ and \x4,x5,x6\, of disjoint triangles and a cocircuit, \x2,x3,x4,x5\, where M\backslash x3 has an N-minor and is \ffsc. We also showed that, when M has a good bowtie, either M\backslash x3,x6 has an N-minor and M\backslash x6 is (4,4,S)-connected; or M\backslash x3/x2 has an N-minor and is \ffsc. In this paper, we show that, when M\backslash x3,x6 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 the penultimate step towards obtaining a splitter theorem for the class of internally 4-connected binary matroids.