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A parallel minor is obtained from a graph by any sequence of edge contractions and parallel edge deletions. We prove that, for any positive integer k, every internally 4-connected graph of sufficiently high order contains a parallel minor isomorphic to a variation of K 4,k with a complete graph on the vertices of degree k, the k-partition triple fan with a… (More)

Let M be a matroid. When M is 3-connected, Tutte's 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… (More)

This paper proves a preliminary step towards a splitter theorem for internally 4-connected binary matroids. In particular, we show that, provided M or M * is not a cubic Möbius or planar ladder or a certain coexten-sion thereof, an internally 4-connected binary matroid M with an internally 4-connected proper minor N either has a proper internally… (More)

In our quest to find a splitter theorem for internally 4-connected binary matroids, we proved in the preceding paper in this series that, except when M or its dual is a cubic Möbius or planar ladder or a certain coextension thereof, an internally 4-connected binary matroid M with an internally 4-connected proper minor N either has a proper internally… (More)

Let M be an internally 4-connected binary matroid and N be an internally 4-connected proper minor of M. In our search for a splitter theorem for internally 4-connected binary matroids, we proved in the third paper in this series that, except when M or its dual is a cubic Möbius or planar ladder or a certain coextension thereof, either M has a proper… (More)

To the memory of Tom Brylawski, who contributed so much to matroid theory. Abstract. We prove that, for each positive integer k, every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M * (K 3,k), M (W k), M (K k), the cycle matroid of the graph obtained from K 2,k by adding paths through the vertices of each vertex class,… (More)

In an earlier paper, we proved that an internally 4-connected binary matroid with at least seven elements contains an internally 4-connected proper minor that is at most six elements smaller. We refine this result, by giving detailed descriptions of the operations required to produce the internally 4-connected minor. Each of these operations is top-down, in… (More)

A graph G is loosely-c-connected, or ℓ-c-connected, if there exists a number d depending on G such that the deletion of fewer than c vertices from G leaves precisely one infinite component and a graph containing at most d vertices. In this paper, we give the structure of a set of ℓ-c-connected infinite graphs that form an unavoidable set among the… (More)

We develop some basic tools to work with representable matroids of bounded tree-width and use them to prove that, for any prime power q and constant k, the characteristic polynomial of any loopless, GF (q)-representable matroid with tree-width k has no real zero greater than q k−1 .