Graphs are to matroids, as embedded graphs are to what?

Iain Moffatt

Abstract

Matroid theory is often thought of as a generalisation of graph theory. Many results in graph theory turn out to be special cases of results in matroid theory. This is beneficial in two ways. Firstly, graph theory can serve as an excellent guide for studying matroids. Secondly,  matroid theory can lead to new, and more general, results about graphs. Thus graph theory and matroid theory are mutually enriching.

In this talk I will be interested in embedded graphs (i.e., graphs in surfaces), rather than abstract graphs. By moving from an embedded graph to a matroid we generally loose all of its topological information. Thus matroids do not appear to provide a `correct' generalisation of embedded graphs. If matroids don't, what do? In this talk I will propose that delta-matroids play the role of matroids in topological graph theory. Delta-matroids were introduced by Bouchet, and arise by relaxing one of the