This thesis is in two parts. The first two chapters deal with infinite matroids and the remaining three chapters with finite matroids. Chapter 1 has two main results. Firstly we prove that there is no duality function on the class of independence spaces which behaves like duality for finite matroids. Secondly we determine the unique class of preindependence spaces which contains the class of independence spaces and is closed under restriction, contraction and the natural duality function. Two other results of this chapter answer questions of Higgs and Welsh on infinite matroids. The main result of Chapter 2 is a counter-example to a 1967 conjecture of Nash-Williams concerning packing disjoint spanning trees in countably infinite graphs. In the third chapter we prove matroid analogues of various bounds on the chromatic number of a graph due to Matula, Szekeres and Wilf, and Brooks. These matroid results sharpen earlier bounds of Heron and Lindstrőm. In the second half of the chapter we answer a number of questions of Mullin and Stanton related to the critical problem for binary matroids. In Chapter 4 we prove the binary case of a matroid conjecture of Welsh which is a natural analogue of Gallai's theorem that the vertex-stability and vertex-covering numbers of a graph G sum to the number of vertices of G. The Tutte polynomial of a graph is known to be related to the bond percolation model on the graph. In Chapter 5 we introduce a basic abstraction of classical percolation theory, namely percolation on clutters. We give new and simpler proofs of several results of Hammersley and bound the percolation probability above and below. One of the main results shows that the theory of the Tutte polynomial cannot be extended from matroids to arbitrary clutters.

This thesis concerns sufficient conditions for a matroid to admit one of two types of structural characterization: a representation over a finite field or a description as a frame matroid. We call a restriction N of a matroid M modular if, for every flat F of M, r_M(F) + r(N) = r_M(F ∩ E(N)) + r_M(F ∪ E(N)). A consequence of a theorem of Seymour is that any 3-connected matroid with a modular U_{2,3}-restriction is binary. We extend this fact to arbitrary finite fields, showing that if N is a modular rank-3 restriction of a vertically 4-connected matroid M, then any representation of N over a finite field extends to a representation of M. We also look at a more general notion of modularity that applies to minors of a matroid, and use it to present conditions for a matroid with a large projective geometry minor to be representable over a finite field. In particular, we show that a 3-connected, representable matroid with a sufficiently large projective geometry over a finite field GF(q) as a minor is either representable over GF(q) or has a U_{2,q^2+1}-minor. A second result of Seymour is that any vertically 4-connected matroid with a modular M(K_4)-restriction is graphic. Geelen, Gerards, and Whittle partially generalized this from M(K_4) to larger frame matroids, showing that any vertically 5-connected, representable matroid with a rank-4 Dowling geometry as a modular restriction is a frame matroid. As with projective geometries, we prove a version of this result for matroids with large Dowling geometries as minors, providing conditions which imply that they are frame matroids.

Certain graphs can be described by the distribution of the edges in its subgraphs. For example, a cycle C is a graph that satisfies |E(H)| |V (H)| < |E(C)| |V (C)| = 1 for all non-trivial subgraphs of C. Similarly, a tree T is a graph that satisfies |E(H)| |V (H)|−1 ≤ |E(T)| |V (T)|−1 = 1 for all non-trivial subgraphs of T. In general, a balanced graph G is a graph such that |E(H)| |V (H)| ≤ |E(G)| |V (G)| and a 1-balanced graph is a graph such that |E(H)| |V (H)|−1 ≤ |E(G)| |V (G)|−1 for all non-trivial subgraphs of G. Apart from these, for integers k and l, graphs G that satisfy the property |E(H)| ≤ k|V (H)| − l for all non-trivial subgraphs H of G play important roles in defining rigid structures. This dissertation is a formal study of a class of density functions that extends the above mentioned ideas. For a rational number r ≤ 1, a graph G is said to be r-balanced if and only if for each non-trivial subgraph H of G, we have |E(H)| |V (H)|−r ≤ |E(G)| |V (G)|−r . For r > 1, similar definitions are given. Weaker forms of r-balanced graphs are defined and the existence of these graphs is discussed. We also define a class of vulnerability measures on graphs similar to the edge-connectivity of graphs and show how it is related to r-balanced graphs. All these definitions are matroidal and the definitions of r-balanced matroids naturally extend the definitions of r-balanced graphs. The vulnerability measures in graphs that we define are ranked and are lesser than the edge-connectivity. Due to the relationship of the r-balanced graphs with the vulnerability measures defined in the dissertation, identifying r-balanced graphs and calculating the vulnerability measures in graphs prove to be useful in the area of network survivability. Relationships between the various classes of r-balanced matroids and their weak forms are discussed. For r ∈ {0, 1}, we give a method to construct big r-balanced graphs from small r-balanced graphs. This construction is a generalization of the construction of Cartesian product of two graphs. We present an algorithmic solution of the problem of transforming any given graph into a 1-balanced graph on the same number of vertices and edges as the given graph. This result is extended to a density function defined on the power set of any set E via a pair of matroid rank functions defined on the power set of E. Many interesting results may be derived in the future by choosing suitable pairs of matroid rank functions and applying the above result.

This thesis concerns sufficient conditions for a matroid to admit one of two types of structural characterization: a representation over a finite field or a description as a frame matroid. We call a restriction N of a matroid M modular if, for every flat F of M, r_M(F) + r(N) = r_M(F ∩ E(N)) + r_M(F ∪ E(N)). A consequence of a theorem of Seymour is that any 3-connected matroid with a modular U_{2,3}-restriction is binary. We extend this fact to arbitrary finite fields, showing that if N is a modular rank-3 restriction of a vertically 4-connected matroid M, then any representation of N over a finite field extends to a representation of M. We also look at a more general notion of modularity that applies to minors of a matroid, and use it to present conditions for a matroid with a large projective geometry minor to be representable over a finite field. In particular, we show that a 3-connected, representable matroid with a sufficiently large projective geometry over a finite field GF(q) as a minor is either representable over GF(q) or has a U_{2,q^2+1}-minor. A second result of Seymour is that any vertically 4-connected matroid with a modular M(K_4)-restriction is graphic. Geelen, Gerards, and Whittle partially generalized this from M(K_4) to larger frame matroids, showing that any vertically 5-connected, representable matroid with a rank-4 Dowling geometry as a modular restriction is a frame matroid. As with projective geometries, we prove a version of this result for matroids with large Dowling geometries as minors, providing conditions which imply that they are frame matroids.

Thesis (Ph. D.)--West Virginia University, 2008. / Title from document title page. Document formatted into pages; contains vi, 44 p. : ill. Includes abstract. Includes bibliographical references (p. 42-44).

Thesis (M.Sc.)--Victoria University of Wellington, 2008. / Includes bibliographical references and index.

Thesis (Ph. D.)--West Virginia University, 2007. / Title from document title page. Document formatted into pages; contains vi, 49 p. : ill. Includes abstract. Includes bibliographical references (p. 45-49).

Matorids provide useful abstraction in combinatorics and have a number of applications in many areas. Gammoids,  which is one of many classes of matroids, and they can be represented by directed graphs, which make them easy to visualize. Due to matroids being discovered quite a long time ago, there are a number of great papers and books to do research on. From results made by Albrecht Immanuel and others, it is made clear that by transforming a gammoid into its standard representation, the cyclic flats can be found via its dual representation. Based on his results, it is possible to find the cyclic flats of any gammoid by finding the dual.

Thesis. 1978. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Bibliography: leaves 132-137. / Ph.D.

This thesis aims to extend some of the results of the Graph Minors Project of Robertson and Seymour to "group-labelled graphs". Let $\Gamma$ be a group. A $\Gamma$-labelled graph is an oriented graph with its edges labelled from $\Gamma$, and is thus a generalization of a signed graph. Our primary result is a generalization of the main result from Graph Minors XIII. For any finite abelian group $\Gamma$, and any fixed $\Gamma$-labelled graph $H$, we present a polynomial-time algorithm that determines if an input $\Gamma$-labelled graph $G$ has an $H$-minor. The correctness of our algorithm relies on much of the machinery developed throughout the graph minors papers. We therefore hope it can serve as a reasonable introduction to the subject. Remarkably, Robertson and Seymour also prove that for any sequence $G_1, G_2, \dots$ of graphs, there exist indices $i<j$ such that $G_i$ is isomorphic to a minor of $G_j$. Geelen, Gerards and Whittle recently announced a proof of the analogous result for $\Gamma$-labelled graphs, for $\Gamma$ finite abelian. Together with the main result of this thesis, this implies that membership in any minor closed class of $\Gamma$-labelled graphs can be decided in polynomial-time. This also has some implications for well-quasi-ordering certain classes of matroids, which we discuss.