Disasters in Abstracting Combinatorial Properties of Linear Dependence

A notion of geometric structure can be given to a set of points without using a coordinate system by instead describing geometric relations between finite combinations of elements. The fundamental problem is to then characterize when the points of such a “geometry” have a consistent coordinatization. Matroids are a first step in such a characterization as they require that geometric relations satisfy inherent abstract properties. Concretely, let E be a finite set and I be a collection of subsets of E. The problem is to characterize pairs (E,I) for which there exists a “representation” of E as vectors in a vector space over a field F where I corresponds to the linear independent subsets of E. Necessary conditions for such a representation to exist include: the empty set is independent, subsets of independent sets are also independent, and for each subset X, the maximal independent subsets of X have the same size. When these properties hold, we say that (E,I) describes a matroid. As a result of these properties, matroids provide many useful concepts and are an appropriate context in which to consider characterizations. Mayhew, Newman, and Whittle showed that there exist pathological obstructions to natural axiomatic and forbidden-substructure characterizations of real-representable matroids. Furthermore, an extension of a result of Seymour illustrates that there is high computational complexity in verifying that a representation exists. This thesis shows that such pathologies still persist even if it is known that there exists a coordinatization with complex numbers and a sense of orientation, both of which are necessary to have a coordinatization over the reals

The circuit and cocircuit lattices of a regular matroid

A matroid abstracts the notions of dependence common to linear algebra, graph theory, and geometry. We show the equivalence of some of the various axiom systems which define a matroid and examine the concepts of matroid minors and duality before moving on to those matroids which can be represented by a matrix over any field, known as regular matroids. Placing an orientation on a regular matroid allows us to define certain lattices (discrete groups) associated to the matroid. These allow us to construct the Jacobian group of a regular matroid analogous to the Jacobian group of a graph. We then survey some recent work characterizing the matroid Jacobian. Finally we extend some results due to Eppstein concerning the Jacobian group of a graph to the case of regular matroids

On the interplay between embedded graphs and delta-matroids

The mutually enriching relationship between graphs and matroids has motivated discoveries in both fields. In this paper, we exploit the similar relationship between embedded graphs and delta-matroids. There are well-known connections between geometric duals of plane graphs and duals of matroids. We obtain analogous connections for various types of duality in the literature for graphs in surfaces of higher genus and delta-matroids. Using this interplay, we establish a rough structure theorem for delta-matroids that are twists of matroids, we translate Petrie duality on ribbon graphs to loop complementation on delta-matroids, and we prove that ribbon graph polynomials, such as the Penrose polynomial, the characteristic polynomial, and the transition polynomial, are in fact delta-matroidal. We also express the Penrose polynomial as a sum of characteristic polynomials

Inductive tools for connected ribbon graphs, delta-matroids and multimatroids

We prove a splitter theorem for tight multimatroids, generalizing the corresponding result for matroids, obtained independently by Brylawski and Seymour. Further corollaries give splitter theorems for delta-matroids and ribbon graphs

Inductive tools for connected delta-matroids and multimatroids

We prove a splitter theorem for tight multimatroids, generalizing the corresponding result for matroids, obtained independently by Brylawski and Seymour. Further corollaries give splitter theorems for delta-matroids and ribbon graphs

Matroids, delta-matroids and embedded graphs

Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic matroids to the setting of embedded graphs. We show that various basic ribbon graph operations and concepts have delta-matroid analogues, and illustrate how the connections between embedded graphs and delta-matroids can be exploited. Also, in direct analogy with the fact that the Tutte polynomial is matroidal, we show that several polynomials of embedded graphs from the literature, including the Las Vergnas, Bollabás-Riordan and Krushkal polynomials, are in fact delta-matroidal