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

Graph Theory

[no abstract available

Some classes of graphs that are nearly cycle-free

A graph is almost series-parallel if there is some edge that one can add to the graph and then contract out to leave a series-parallel graph, that is, a graph with no K4-minor. In this dissertation, we find the full list of excluded minors for the class of graphs that are almost series-parallel. We also obtain the corresponding result for the class of graphs such that uncontracting an edge and then deleting the uncontracted edge produces a series-parallel graph. A notable feature of a 3-connected almost series-parallel graph is that it has two vertices whose removal leaves a tree. This motivates consideration of those graphs for which there are two vertices whose removal is cycle-free. We find the full list of excluded minors for the class of graphs that have a set of at most two vertices whose removal is cycle-free

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