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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)

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)

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)

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)

If S is a set of matroids, then the matroid M is S-fragile if, for every element e ∈ E(M), either M \e or M/e has no minor isomorphic to a member of S. Excluded-minor characterizations often depend, implicitly or explicitly, on understanding classes of fragile matroids. In certain cases, when M is a minor-closed class of S-fragile matroids, and N ∈ M, the… (More)

We characterize all internally 4-connected binary matroids M with the property that the ground set of M can be ordered ei+t} is 4-separating for all 0 ≤ i, t ≤ n − 1 (all subscripts are read modulo n). We prove that in this case either n ≤ 7 or, up to duality, M is isomorphic to the polygon matroid of a cubic or quartic planar ladder, the polygon matroid of… (More)

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