Polytopal and structural aspects of matroids and related objects
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This thesis consists of three self-contained but related parts. The rst is focussed on polymatroids, these being a natural generalisation of matroids. The Tutte polynomial is one of the most important and well-known graph polynomials, and also features prominently in matroid theory. It is however not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. The second section is concerned with split matroids, a class of matroids which arises by putting conditions on the system of split hyperplanes of the matroid base polytope. We describe these conditions in terms of structural properties of the matroid, and use this to give an excluded minor characterisation of the class. In the nal section, we investigate the structure of clutters. A clutter consists of a nite set and a collection of pairwise incomparable subsets. Clutters are natural generalisations of matroids, and they have similar operations of deletion and contraction. We introduce a notion of connectivity for clutters that generalises that of connectivity for matroids. We prove a splitter theorem for connected clutters that has the splitter theorem for connected matroids as a special case: if M and N are connected clutters, and N is a proper minor of M, then there is an element in E(M) that can be deleted or contracted to produce a connected clutter with N as a minor.
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