Infinite graphs, graph-like spaces and B-matroids

The central theme of this thesis is to prove results about infinite mathematical objects by studying the behaviour of their finite substructures. In particular, we study B-matroids, which are an infinite generalization of matroids introduced by Higgs \cite{higgs}, and graph-like spaces, which are topological spaces resembling graphs, introduced by Thomassen and Vella \cite{thomassenvella}. Recall that the circuit matroid of a finite graph is a matroid defined on the edges of the graph, with a set of edges being independent if it contains no circuit. It turns out that graph-like continua and infinite graphs both have circuit B-matroids. The first main result of this thesis is a generalization of Whitney's Theorem that a graph has an abstract dual if and only if it is planar. We show that an infinite graph has an abstract dual (which is a graph-like continuum) if and only if it is planar, and also that a graph-like continuum has an abstract dual (which is an infinite graph) if and only if it is planar. This generalizes theorems of Thomassen (\cite{thomassendual}) and Bruhn and Diestel (\cite{bruhndiestel}). The difficult part of the proof is extending Tutte's characterization of graphic matroids (\cite{tutte2}) to finitary or co-finitary B-matroids. In order to prove this characterization, we introduce a technique for obtaining these B-matroids as the limit of a sequence of finite minors. In \cite{tutte}, Tutte proved important theorems about the peripheral (induced and non-separating) circuits of a $3$-connected graph. He showed that for any two edges of a $3$-connected graph there is a peripheral circuit containing one but not the other, and that the peripheral circuits of a $3$-connected graph generate its cycle space. These theorems were generalized to $3$-connected binary matroids by Bixby and Cunningham (\cite{bixbycunningham}). We generalize both of these theorems to $3$-connected binary co-finitary B-matroids. Richter, Rooney and Thomassen \cite{richterrooneythomassen} showed that a locally connected, compact metric space has an embedding in the sphere unless it contains a subspace homeomorphic to $K_5$ or $K_{3,3}$, or one of a small number of other obstructions. We are able to extend this result to an arbitrary surface $\Sigma$; a locally connected, compact metric space embeds in $\Sigma$ unless it contains a subspace homeomorphic to a finite graph which does not embed in $\Sigma$, or one of a small number of other obstructions

On graph-like continua of finite length

We extend the notion of effective resistance to metric spaces that are similar to graphs but can also be similar to fractals. Combined with other basic facts proved in the paper, this lays the ground for a construction of Brownian Motion on such spaces completed in [10]

Infinite matroids in graphs

It has recently been shown that infinite matroids can be axiomatized in a way that is very similar to finite matroids and permits duality. This was previously thought impossible, since finitary infinite matroids must have non-finitary duals. In this paper we illustrate the new theory by exhibiting its implications for the cycle and bond matroids of infinite graphs. We also describe their algebraic cycle matroids, those whose circuits are the finite cycles and double rays, and determine their duals. Finally, we give a sufficient condition for a matroid to be representable in a sense adapted to infinite matroids. Which graphic matroids are representable in this sense remains an open question.Comment: Figure correcte

Infinite graphic matroids Part I

An infinite matroid is graphic if all of its finite minors are graphic and the intersection of any circuit with any cocircuit is finite. We show that a matroid is graphic if and only if it can be represented by a graph-like topological space: that is, a graph-like space in the sense of Thomassen and Vella. This extends Tutte's characterization of finite graphic matroids. The representation we construct has many pleasant topological properties. Working in the representing space, we prove that any circuit in a 3-connected graphic matroid is countable

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

MacLane's Theorem for Graph-Like Spaces

The cycle space of a finite graph is the subspace of the edge space generated by the edge sets of cycles, and is a well-studied object in graph theory. Recently progress has been made towards extending the theory of cycle spaces to infinite graphs. Graph-like spaces are a class of topological objects that reconcile the combinatorial properties of infinite graphs with the topological properties of finite graphs. They were first introduced by Thomassen and Vella as a natural, general class of topological spaces for which Menger's Theorem holds. Graph-like spaces are the natural objects for extending classical results from topological graph theory and cycle space theory to infinite graphs. This thesis focuses on the topological properties of embeddings of graph-like spaces, as well as the algebraic properties of graph-like spaces. We develop a theory of embeddings of graph-like spaces in surfaces. We also show how the theory of edge spaces developed by Vella and Richter applies to graph-like spaces. We combine the topological and algebraic properties of embeddings of graph-like spaces in order to prove an extension of MacLane's Theorem. We also extend Thomassen's version of Kuratowski's Theorem for 2-connected compact locally connected metric spaces to the class of graph-like spaces