The Smallest Rounded Sets of Binary Matroids

It was proved by Oxley that U2,4 is the only non-trivial 3-connected matroid N such that, whenever a 3-connected matroid M has an N-minor and x and y are elements of M, there is an N-minor of M using {x, y} . This paper establishes the corresponding result for binary matroids by proving that if M and N above must both be binary, then there are exactly two possibilities for N: the rank-three and rank-four wheels. © 1990, Academic Press Limited. All rights reserved

Triangle-roundedness in matroids

A matroid $N$ is said to be triangle-rounded in a class of matroids $\mathcal{M}$ if each $3$-connected matroid $M\in \mathcal{M}$ with a triangle $T$ and an $N$-minor has an $N$-minor with $T$ as triangle. Reid gave a result useful to identify such matroids as stated next: suppose that $M$ is a binary $3$-connected matroid with a $3$-connected minor $N$, $T$ is a triangle of $M$ and $e\in T\cap E(N)$; then $M$ has a $3$-connected minor $M'$ with an $N$-minor such that $T$ is a triangle of $M'$ and $|E(M')|\le |E(N)|+2$. We strengthen this result by dropping the condition that such element $e$ exists and proving that there is a $3$-connected minor $M'$ of $M$ with an $N$-minor $N'$ such that $T$ is a triangle of $M'$ and $E(M')-E(N')\subseteq T$. This result is extended to the non-binary case and, as an application, we prove that $M(K_5)$ is triangle-rounded in the class of the regular matroids

On Minors Avoiding Elements in Matroids

Let ℱ be a collection of 3-connected matroids, none a proper minor of another, such that if M is a 3-connected matroid having a proper ℱ-minor and e is an element of M, then M has an ℱ-minor avoiding e. This paper proves that there are precisely two collections ℱ with this property: {U2,4} and {U2,4, M(K4)}. Several extensions of this result and some similar results for 2-connected matroids are also established. © 1991, Academic Press Limited. All rights reserved

Triangles in 3-connected matroids

AbstractA collection F of 3-connected matroids is triangle-rounded if, whenever M is a 3-connected matroid having a minor in F, and T is a 3-element circuit of M, then M has a minor which uses T and is isomorphic to a member of F. An efficient theorem for testing a collection of matroids for this property is presented. This test is used to obtain several results including the following extension of a result of Asano, Nishizeki, and Seymour. Let T be a 3-element circuit of a 3-connected binary nonregular matroid M with at least eight elements. Then M has a minor using T that is isomorphic to S8 or the generalized parallel connection across T of F7 and M(K4)

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

Capturing two elements in unavoidable minors of 3-connected binary matroids

Let M be a 3-connected binary matroid and let n be an integer exceeding 2. Ding, Oporowski, Oxley, and Vertigan proved that there is an integer f(n) so that if |E(M)|\u3ef(n), then M has a minor isomorphic to one of the rank-n wheel, the rank-n tipless binary spike, or the cycle or bond matroid of K3 n. This result was recently extended by Chun, Oxley, and Whittle to show that there is an integer g(n) so that if |E(M)|\u3eg(n) and xεE(M), then x is an element of a minor of M isomorphic to one of the rank-n wheel, the rank-n binary spike with a tip and a cotip, or the cycle or bond matroid of K11,1,n. In this paper, we prove that, for each i in {2,3}, there is an integer hi(n) so that if |E(M)|\u3ehi(n) and Z is an i-element rank-2 subset of M, then M has a minor from the last list whose ground set contains Z. © 2012 Elsevier Inc

Totally free expansions of matroids

The aim of this paper is to give insight into the behaviour of inequivalent representations of 3-connected matroids. An element x of a matroid M is fixed if there is no extension M′ of M by an element x′ such that {x, x′} is independent and M′ is unaltered by swapping the labels on x and x′. When x is fixed, a representation of M.\x extends in at most one way to a representation of M. A 3-connected matroid N is totally free if neither N nor its dual has a fixed element whose deletion is a series extension of a 3-connected matroid. The significance of such matroids derives from the theorem, established here, that the number of inequivalent representations of a 3-connected matroid M over a finite field F is bounded above by the maximum, over all totally free minors N of M, of the number of inequivalent F-representations of N. It is proved that, within a class of matroids that is closed under minors and duality, the totally free matroids can be found by an inductive search. Such a search is employed to show that, for all r ≥ 4, there are unique and easily described rank-r quaternary and quinternary matroids, the first being the free spike. Finally, Seymour\u27s Splitter Theorem is extended by showing that the sequence of 3-connected matroids from a matroid M to a minor N, whose existence is guaranteed by the theorem, may be chosen so that all deletions and contractions of fixed and cofixed elements occur in the initial segment of the sequence. © 2001 Elsevier Science

Capturing elements in matroid minors

In this dissertation, we begin with an introduction to a matroid as the natural generalization of independence arising in three different fields of mathematics. In the first chapter, we develop graph theory and matroid theory terminology necessary to the topic of this dissertation. In Chapter 2 and Chapter 3, we prove two main results. A result of Ding, Oporowski, Oxley, and Vertigan reveals that a large 3-connected matroid M has unavoidable structure. For every n exceeding two, there is an integer f(n) so that if |E(M)| exceeds f(n), then M has a minor isomorphic to the rank-n wheel or whirl, a rank-n spike, the cycle or bond matroid of K_{3,n}, or U_{2,n} or U_{n-2,n}. In Chapter 2, we build on this result to determine what can be said about a large structure using a specified element e of M. In particular, we prove that, for every integer n exceeding two, there is an integer g(n) so that if |E(M)| exceeds g(n), then e is an element of a minor of M isomorphic to the rank-n wheel or whirl, a rank-n spike, the cycle or bond matroid of K_{1,1,1,n}, a specific single-element extension of M(K_{3,n}) or the dual of this extension, or U_{2,n} or U_{n-2,n}. In Chapter 3, we consider a large 3-connected binary matroid with a specified pair of elements. We extend a corollary of the result of Chapter 2 to show the following result for any pair {x,y} of elements of a 3-connected binary matroid M. For every integer n exceeding two, there is an integer h(n) so that if |E(M)| exceeds h(n), then x and y are elements of a minor of M isomorphic to the rank-n wheel, a rank-n binary spike with a tip and a cotip, or the cycle or bond matroid of K_{1,1,1,n}

Structure and Minors in Graphs and Matroids.

This dissertation establishes a number of theorems related to the structure of graphs and, more generally, matroids. In Chapter 2, we prove that a 3-connected graph G that has a triangle in which every single-edge contraction is 3-connected has a minor that uses the triangle and is isomorphic to K5 or the octahedron. We subsequently extend this result to the more general context of matroids. In Chapter 3, we specifically consider the triangle-rounded property that emerges in the results of Chapter 2. In particular, Asano, Nishizeki, and Seymour showed that whenever a 3-connected matroid M has a four-point-line-minor, and T is a triangle of M, there is a four-point-line-minor of M using T. We will prove that the four-point line is the only such matroid on at least four elements. In Chapter 4, we extend a result of Dirac which states that any set of n vertices of an n-connected graph lies in a cycle. We prove that if V\u27 is a set of at most 2n vertices in an n-connected graph G, then G has, as a minor, a cycle using all of the vertices of V\u27. In Chapter 5, we prove that, for any vertex v of an n-connected simple graph G, there is a n-spoked-wheel-minor of G using v and any n edges incident with v. We strengthen this result in the context of 4-connected graphs by proving that, for any vertex v of a 4-connected simple graph G, there is a K 5- or octahedron-minor of G using v and any four edges incident with v. Motivated by the results of Chapters 4 and 5, in Chapter 6, we introduce the concept of vertex-roundedness. Specifically, we provide a finite list of conditions under which one can determine which collections of graphs have the property that whenever a sufficiently highly connected graph has a minor in the collection, it has such a minor using any set of vertices of some fixed size

On Roundedness in Matroid Theory.

This thesis studies the relationship between subsets and specified minors in a 3-connected matroid. For positive integers k and m, a set S of k-connected matroids is (k,m)-rounded if it satisfies the following condition. Whenever M is a k-connected matroid having an S-minor and X is a subset of E(M) with at most m elements, then M has an S-minor using X. Oxley characterized the (3,2)-rounded sets that contain a single matroid. In Chapter 2, we obtain an analog of this result for binary matroids. In Chapter 3, we use this result to characterize the pairs of matroids which form (3,2)-rounded sets. The methods of Chapter 3 are generalized to 4-connected matroids in Chapter 4 to determine the (4,2)-rounded sets that contain a single matroid. This extends results of Coullard and Kahn. For a 3-connected minor N of a 3-connected matroid M, the following question arises from roundedness theory. Let X be a subset of E(M). How small a 3-connected minor of M can we find which both uses X and has an N-minor? Seymour answered this question for $\vert$X$\vert$ = 1 and 2. We answer this question for $\vert$X$\vert$ $\geq$ 3 in Chapter 5. Finally, in Chapter 6, results from roundedness theory are applied to the study of 3-element circuits in 3-connected matroids. An extension of a result of Asano, Nishizeki, and Seymour is obtained for binary matroids which are non-regular