The regular matroids with no 5-wheel minor

For r in {3, 4}, the class of binary matroids with no minor isomorphic to M(Wr), the rank-r wheel, has an easily described structure. This paper determines all graphs with no W5-minor and uses this to show that the class of regular matroids with no M(W5)-minor also has a relatively simple structure. © 1989

A Characterization of Certain Excluded-Minor Classes of Matroids

A result of Walton and the author establishes that every 3-connected matroid of rank and corank at least three has one of five six-element rank-3 self-dual matroids as a minor. This paper characterizes two classes of matroids that arise when one excludes as minors three of these five matroids. One of these results extends the author\u27s characterization of the ternary matroids with no M(K4)-minor, while the other extends Tutte\u27s excluded-minor characterization of binary matroids. © 1989, Academic Press Limited. All rights reserved

On an excluded-minor class of matroids

A result of Walton and the author establishes that every 3-connected matroid of rank and corank at least three has one of five 6-element rank-3 self-dual matroids as a minor. One of these matroids is the rank-3 whirl W3. Another is the rank-3 matroid P6 that consists of a single 3-point line together with three points off the line. This paper determines the structure of the class of matroids that is obtained by excluding as minors both W3 and P6. As a consequence of this result, we deduce a characterization of the class of GF(4)-representable matroids with no W3-minor. © 1990

On the Structure of 3-connected Matroids and Graphs

An element e of a 3-connected matroid M is essential if neither the deletion M\e nor the contraction M/e is 3-connected. Tutte\u27s Wheels and Whirls Theorem proves that the only 3-connected matroids in which every element is essential are the wheels and whirls. In this paper, we consider those 3-connected matroids that have some non-essential elements, showing that every such matroid M must have at least two such elements. We prove that the essential elements of M can be partitioned into classes where two elements are in the same class if M has a fan, a maximal partial wheel, containing both. We also prove that if an essential element e of M is in more than one fan, then that fan has three or five elements; in the latter case, e is in exactly three fans. Moreover, we show that if M has a fan with 2k or 2k + 1 elements for some k ≥ 2, then M can be obtained by sticking together a (k + 1)-spoked wheel and a certain 3-connected minor of M. The results proved here will be used elsewhere to completely determine all 3-connected matroids with exactly two non-essential elements. © 2000 Academic Press

Excluding a small minor

There are sixteen 3-connected graphs on eleven or fewer edges. For each of these graphs H we discuss the structure of graphs that do not contain a minor isomorphic to H. © 2012 Elsevier B.V. All rights reserved

Constructing internally 4-connected binary matroids

This is the post-print version of the Article - Copyright @ 2013 ElsevierIn 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 that it produces a smaller minor from the original. We also describe each as a bottom-up operation, constructing a larger matroid from the original, and we give necessary and su fficient conditions for each of these bottom-up moves to produce an internally 4-connected binary matroid. From this, we derive a constructive method for generating all internally 4-connected binary matroids.This study is supported by NSF IRFP Grant 0967050, the Marsden Fund, and the National Security Agency

A chain theorem for internally 4-connected binary matroids

This is the post-print version of the Article - Copyright @ 2011 ElsevierLet 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 proper minor N with |E(M) − E(N)| at most 3 unless M or its dual is the cycle matroid of a planar or Möbius quartic ladder, or a 16-element variant of such a planar ladder.This study was partially supported by the National Security Agency