## LSU Historical Dissertations and Theses

#### Title

Structure and Minors in Graphs and Matroids.

1999

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

James G. Oxley

#### Abstract

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' 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'. 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.

9780599474765

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