Date of Award
Master of Arts in Mathematics
First Reader/Committee Chair
Dr. Aikin, Jeremy
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.
Vega, Vanessa Natalie, "Excluded minors for nearly-paving matroids" (2020). Electronic Theses, Projects, and Dissertations. 1130.