Computing excluded minors for classes of matroids representable over partial fields

We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids.We describe an implementation of a computer search for the "small" excluded minors for a class of matroids representable over a partial field. Using these techniques, we enumerate the excluded minors on at most 15 elements for both the class of dyadic matroids, and the class of 2-regular matroids. We conjecture that there are no other excluded minors for the class of 2-regular matroids; whereas, on the other hand, we show that there is a 16-element excluded minor for the class of dyadic matroids

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

The excluded minors for the class of matroids that are binary or ternary

We show that the excluded minors for the class of matroids that are binary or ternary are U2,5, U3,5, U2,4position indicatorF7, U2,4position indicatorF7*, U2,4position indicator2F7, U2,4position indicator2F7*, and the unique matroids obtained by relaxing a circuit-hyperplane in either AG(3,2) or T12. The proof makes essential use of results obtained by Truemper on the structure of almost-regular matroids. © 2011 Elsevier Ltd

Selected Problems on Matroid Minors

This dissertation begins with an introduction to matroids and graphs. In the first chapter, we develop matroid and graph theory definitions and preliminary results sufficient to discuss the problems presented in the later chapters. These topics include duality, connectivity, matroid minors, and Cunningham and Edmonds\u27s tree decomposition for connected matroids. One of the most well-known excluded-minor results in matroid theory is Tutte\u27s characterization of binary matroids. The class of binary matroids is one of the most widely studied classes of matroids, and its members have many attractive qualities. This motivates the study of matroid classes that are close to being binary. One very natural such minor-closed class Z consists of those matroids M such that the deletion or the contraction of e is binary for all elements e of M. Chapter 2 is devoted to determining the set of excluded minors for Z. Duality plays a central role in the study of matroids. It is therefore natural to ask the following question: which matroids guarantee that, when present as minors, their duals are present as minors? We answer this question in Chapter 3. We also consider this problem with additional constraints regarding the connectivity and representability of the matroids in question. The main results of Chapter 3 deal with 3-connected matroids

Detachable Pairs in 3-Connected Matroids

The classical tool at the matroid theorist’s disposal when dealing with the common problem of wanting to remove a single element from a 3-connected matroid without losing 3-connectivity is Tutte’s Wheels-and-Whirls Theorem. However, situations arise where one wishes to delete or contract a pair of elements from a 3-connected matroid whilst maintaining 3-connectedness. The goal of this research was to provide a new tool for making such arguments. Let M be a 3-connected matroid. A detachable pair in M is a pair x, y ∈ E(M) such that either M\x, y or M/x, y is 3-connected. Naturally, our aim was to find the necessary conditions on M which guarantee the existence of a detachable pair. Triangles and triads are an obvious barrier to overcome, and can be done so by allowing the use of a Δ − Y exchange. Apart from these matroids with three-element 3-separating sets, the only other class of matroids that fail to contain a detachable pair for which no bound can be placed on the size of the ground set is the class of spikes. In particular, we prove the following result. Let M be a 3-connected matroid with at least thirteen elements. If M is not a spike, then either M contains a detachable pair, or there exists a matroid M′ where M′ is obtained by performing a single Δ − Y exchange on either M or M* such that M′ contains a detachable pair. As well as being an important theorem in its own right, we anticipate that this result will be essential in future attempts to extend Seymour’s Splitter Theorem in a comparable manner; where the goal would be to obtain a detachable pair as well as maintaining a 3-connected minor. As such, much work has been done herein to study the precise configurations that arise in 3-separating subsets which themselves yield no detachable pair

Connectivity for Matroids and Graphs.

This dissertation studies connectivity for matroids and graphs. The main results generalize Tutte\u27s Wheels and Whirls Theorem and have numerous applications. In Chapter 2, we prove two structural theorems for 3-connected matroids. 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. If M is not a wheel or a whirl, 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 containing both. In particular, M must have at least two non-essential elements. In the second structural theorem, we show that if M has a fan with 2k or 2k + 1 elements for some $k \geq \ 2$, then M can be obtained by sticking together a (k + 1)-spoked wheel and a certain 3-connected minor of M. In Chapters 3 and 4, we characterize all 3-connected matroids whose set of non-essential elements has rank two. In particular, we completely determine all 3-connected matroids with exactly two non-essential elements. In Chapter 5, we derive some consequences of these results for the 3-connected binary matroids and graphs. We prove that there are exactly six classes of 3-connected binary matroids whose set of non-essential elements has rank two and we prove that there are exactly two classes of graphs, multi-dimensional wheels and twisted wheels, with exactly two non-essential edges. In Chapter 6, we use our first structural theorem to investigate the set of elements e in a 3-connected matroid M such that the simplification of M/e is 3-connected. We get best-possible lower bounds on the number of such elements thereby improving a result which was derived by Cunningham and Seymour independently. We also give some generalizations of the Wheels and Whirls Theorem and the Wheels Theorem