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

Let M be a 3-connected binary matroid; M is called internally 4-connected if one side of every 3-separation is a triangle or a triad, and M is internally 4-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\x3 has an N-minor and is internally 4-connected. We also showed that, when M has a good bowtie, either M\x3,x6 has an N-minor; or M\x3/x2 has an N-minor and is internally 4-connected. In this paper, we show that, when M\x3,x6 has an N-minor but is not internally 4-connected, 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 one of several special substructures of M. This is a significant step towards obtaining a splitter theorem for the class of internally 4-connected binary matroids

Towards a splitter theorem for internally 4-connected binary matroids VIII: small matroids

Our splitter theorem for internally 4-connected binary matroids studies pairs of the form (M,N), where N and M are internally 4-connected binary matroids, M has a proper N-minor, and if M' is an internally 4-connected matroid such that M has a proper M'-minor and M' has an N-minor, then |E(M)|-|E(M')|>3. The analysis in the splitter theorem requires the constraint that |E(M)|>15. In this article, we complement that analysis by using an exhaustive computer search to find all such pairs satisfying |E(M)|<16.Comment: Correcting minor error

Towards a splitter theorem for internally 4-connected binary matroids IX: The theorem

Let M be a binary matroid that is internally 4-connected, that is, M is 3-connected, and one side of every 3-separation is a triangle or a triad. Let N be an internally 4-connected proper minor of M. In this paper, we 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 three elements, or by removing some set of elements in an easily described way from one of a small collection of special substructures of M