Two generalisations of the wheels-and-whirls theorem.

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Theses / Dissertations
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Doctor of Philosophy
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Toft, Gerry

One of the most famous results in matroid theory is Tutte’s Wheels-and-Whirls Theorem. It states that every 3-connected matroid has an element which can either be deleted or con- tracted while retaining 3-connectivity, except for two families of matroids: the eponymous wheels and whirls. The Wheels-and-Whirls Theorem is a powerful tool for inductive argu- ments on 3-connected matroids. We consider two generalisations of the Wheels-and-Whirls Theorem.

First, what are the k-connected matroids such that the deletion and contraction of every element is not k-connected? Motivated by this problem, we consider matroids in which every element is contained in a small circuit and a small cocircuit, and, in particular, when these circuits and cocircuits have a cyclic structure. The first part of this thesis is concerned with matroids in which have a cyclic ordering σ of their ground set such that every set of s − 1 consecutive elements of σ is contained in an s-element circuit and every set of t − 1 consecutive elements of σ is contained in a t-element circuit. We show that these matroids are highly structured by proving that they are “(s, t)-cyclic”, that is, their s-element circuits and t-element cocircuits are consecutive in σ in a prescribed way. Next, we provide a characterisation of these matroids by showing that every (s, t)-cyclic matroid is a weak-map image of a particular (s, t)-cyclic matroid.

Secondly, what are the 3-connected matroids such that such that the deletion and con- traction of every 2-element subset is not 3-connected? In the second part of this thesis, we find all such matroids. Roughly speaking, these matroids can be constructed in one of four ways: by attaching fans to a spike, by attaching fans to a line, by attaching particular matroids to M (K3,m), or by attaching particular matroids to each end of a fan.

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