Aspects of matroid connectivity and uniformity. (2022)
Type of ContentTheses / Dissertations
Degree NameDoctor of Philosophy
In approaching a combinatorial problem, it is often desirable to be armed with a notion asserting that some objects are more highly structured than others. In particular, focusing on highly structured objects may avoid certain degeneracies and allow for the core of the problem to be addressed. In matroid theory, the principle notion fulfilling this role of “structure” is that of connectivity. This thesis proves a number of results furthering the knowledge of matroid connectivity and also introduces a new structural notion, that of generalised uniformity.
The first part of this thesis considers 3-connected matroids and the presence of elements which may be deleted or contracted without the introduction of any non-minimal 2-separations. Principally, a Wheels-and-Whirls Theorem and then a Splitter Theorem is established, guaranteeing the existence of such elements, provided certain well-behaved structures are not present.
The second part of this thesis generalises the notion of a uniform matroid by way of a 2-parameter property capturing “how uniform” a given matroid is. Initially, attention is focused on matroids representable over some field. In particular, a finiteness result is established and a specific class of binary matroids is completely determined. The concept of generalised uniformity is then considered more broadly by an analysis of its relevance to a number of established matroid notions and settings. Within that analysis, a number of equivalent characterisations of generalised uniformity are obtained.
Lastly, the third part of the thesis considers a highly structured class of matroids whose members are defined by the nature of their circuits. A characterisation is achieved for the regular members of this class and, in general, the infinitely many excluded series minors are determined.
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Brettell, Nicholas John (University of Canterbury. School of Mathematics and Statistics, 2014)Connectivity is a fundamental tool for matroid theorists, which has become increasingly important in the eventual solution of many problems in matroid theory. Loosely speaking, connectivity can be used to help describe a ...
Oxley, James; Semple, Charles; Whittle, Geoff (2009)Let 𝑀 be a 3-connected matroid other than a wheel or a whirl. In the next paper in this series, we prove that there is an element whose deletion from 𝑀 or 𝑀∗ is 3-connected and whose only 3- separations are equivalent ...
Hall, R.; Oxley, J.; Semple, C. (University of Canterbury. Mathematics and Statistics., 2005)Let M be a matroid. When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. This result was extended by Oxley, Semple, and Whittle, who showed that, when M ...