Matroids, Delta-matroids and Embedded Graphs

TL;DR

This paper establishes a new correspondence between embedded graphs and delta-matroids, extending matroid theory to graph embeddings and connecting various graph polynomials to delta-matroids.

Contribution

It introduces the concept that delta-matroids naturally extend graphic matroids to embedded graphs and demonstrates their applications in graph polynomials.

Findings

Delta-matroids arise as extensions of graphic matroids for embedded graphs.

Various ribbon graph operations have delta-matroid analogues.

Several embedded graph polynomials are delta-matroidal.

Abstract

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, Bollabas-Riordan and Krushkal polynomials, are in fact delta-matroidal.