dc.contributor.author	Drummond, George Matthew
dc.date.accessioned	2022-07-31T23:50:45Z
dc.date.available	2022-07-31T23:50:45Z
dc.date.issued	2022	en
dc.description.abstract	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.	en
dc.identifier.uri	https://hdl.handle.net/10092/104032
dc.identifier.uri	http://dx.doi.org/10.26021/13130
dc.language	English
dc.language.iso	en	en
dc.rights	All Right Reserved	en
dc.rights.uri	https://canterbury.libguides.com/rights/theses	en
dc.title	Aspects of matroid connectivity and uniformity.	en
dc.type	Theses / Dissertations	en
thesis.degree.discipline	Mathematics	en
thesis.degree.grantor	University of Canterbury	en
thesis.degree.level	Doctoral	en
thesis.degree.name	Doctor of Philosophy	en
uc.bibnumber	3182124
uc.college	Faculty of Engineering	en

Aspects of matroid connectivity and uniformity.

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