Date of Award
6-2020
Document Type
Thesis
Degree Name
Master of Arts in Mathematics
Department
Mathematics
First Reader/Committee Chair
Dr. Aikin, Jeremy
Abstract
Matroids capture an abstract notion of independence that generalizes linear independence in linear algebra, edge independence in graph theory, as well as algebraic independence. Given a particular property of matroids, all the matroids possessing that property form a matroid class. A common research theme in matroid theory is to characterize matroid classes so that, given a matroid M, it is possible to determine whether or not M belongs to a given class. An excluded minor of a minor-closed class is a matroid N that is, in a sense, minimal with respect to not being in the minor-closed class. An attractive way to characterize a minor-closed class of matroids is to determine the complete list of excluded minors for the minor-closed class. For example, the class of planar graphs is characterized (Kuratowski’s Theorem) by graphs that do not have any minor that is isomorphic to K_5 or K_3,3. In this presentation, we introduce several fundamental minor-closed classes of matroids; namely, uniform matroids and paving matroids. We define closely related minor-closed classes, which we call nearly-uniform and nearly-paving matroids. Finally, we provide an excluded minor characterization for nearly-uniform and nearly-paving matroids.
Recommended Citation
Vega, Vanessa Natalie, "Excluded minors for nearly-paving matroids" (2020). Electronic Theses, Projects, and Dissertations. 1130.
https://scholarworks.lib.csusb.edu/etd/1130