On Binary And Regular Matroids Without Small Minors

The results of this dissertation consist of excluded-minor results for Binary Matroids and excluded-minor results for Regular Matroids. Structural theorems on the relationship between minors and k-sums of matroids are developed here in order to provide some of these characterizations. Chapter 2 of the dissertation contains excluded-minor results for Binary Matroids. The first main result of this dissertation is a characterization of the internally 4-connected binary matroids with no minor that is isomorphic to the cycle matroid of the prism+e graph. This characterization generalizes results of Mayhew and Royle [18] for binary matroids and results of Dirac [8] and Lovász [15] for graphs. The results of this chapter are then extended from the class of internally 4-connected matroids to the class of 3-connected matroids. Chapter 3 of the dissertation contains the second main result, a decomposition theorem for regular matroids without certain minors. This decomposition theorem is used to obtain excluded-minor results for Regular Matroids. Wagner, Lovász, Oxley, Ding, Liu, and others have characterized many classes of graphs that are H-free for graphs H with at most twelve edges (see [7]). We extend several of these excluded-minor characterizations to regular matroids in Chapter 3. We also provide characterizations of regular matroids excluding several graphic matroids such as the octahedron, cube, and the Möbius Ladder on eight vertices. Both theoretical and computer-aided proofs of the results of Chapters 2 and 3 are provided in this dissertation

Splitters and Decomposers for Binary Matroids

Let $EX[M_1\dots, M_k]$ denote the class of binary matroids with no minors isomorphic to $M_1, \dots, M_k$. In this paper we give a decomposition theorem for $EX[S_{10}, S_{10}^*]$, where $S_{10}$ is a certain 10-element rank-4 matroid. As corollaries we obtain decomposition theorems for the classes obtained by excluding the Kuratowski graphs $EX[M(K_{3,3}), M^*(K_{3,3}), M(K_5), M^*(K_5)]$ and $EX[M(K_{3,3}), M^*(K_{3,3})]$. These decomposition theorems imply results on internally $4$-connected matroids by Zhou [\ref{Zhou2004}], Qin and Zhou [\ref{Qin2004}], and Mayhew, Royle and Whitte [\ref{Mayhewsubmitted}].Comment: arXiv admin note: text overlap with arXiv:1403.775

On two classes of nearly binary matroids

We give an excluded-minor characterization for the class of matroids M in which M\e or M/e is binary for all e in E(M). This class is closely related to the class of matroids in which every member is binary or can be obtained from a binary matroid by relaxing a circuit-hyperplane. We also provide an excluded-minor characterization for the second class.Comment: 14 pages, 4 figures. This paper has been accepted for publication in the European Journal of Combinatorics. This is the final version of the pape

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

A decomposition theorem for binary matroids with no prism minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, $K_5\backslash e$. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac's infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connection of $F_7$ with itself across a triangle with an element of the triangle deleted; it's rank is bounded by 5; or it admits a non-minimal exact 3-separation induced by the 3-separation in $P_9$. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, $F_7$ and $PG(3, 2)$, respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of $R_{10}$, the unique splitter for regular matroids. As a corollary, we obtain Dillon, Mayhew, and Royle's result identifying the binary internally 4-connected matroids with no prism minor [5]

A notion of minor-based matroid connectivity

For a matroid $N$, a matroid $M$ is $N$-connected if every two elements of $M$ are in an $N$-minor together. Thus a matroid is connected if and only if it is $U_{1,2}$-connected. This paper proves that $U_{1,2}$ is the only connected matroid $N$ such that if $M$ is $N$-connected with $|E(M)| > |E(N)|$, then $M \backslash e$ or $M / e$ is $N$-connected for all elements $e$. Moreover, we show that $U_{1,2}$ and $M(\mathcal{W}_2)$ are the only connected matroids $N$ such that, whenever a matroid has an $N$-minor using $\{e,f\}$ and an $N$-minor using $\{f,g\}$, it also has an $N$-minor using $\{e,g\}$. Finally, we show that $M$ is $U_{0,1} \oplus U_{1,1}$-connected if and only if every clonal class of $M$ is trivial.Comment: 13 page

Unavoidable parallel minors of regular matroids

This is the post-print version of the Article - Copyright @ 2011 ElsevierWe 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, or the cycle matroid of the graph obtained from K_{3,k} by adding a complete graph on the vertex class with three vertices.This study is partially supported by a grant from the National Security Agency