On the interplay between embedded graphs and delta-matroids

TL;DR

This paper explores the relationship between embedded graphs and delta-matroids, establishing new duality connections, structural theorems, and polynomial invariants that extend classical graph-matroid theory to higher genus surfaces.

Contribution

It introduces a structure theorem for delta-matroids related to matroid twists, links ribbon graph dualities to delta-matroid operations, and shows key ribbon graph polynomials are delta-matroidal.

Findings

Established duality connections for graphs on surfaces of higher genus.

Proved ribbon graph polynomials are delta-matroidal.

Expressed the Penrose polynomial as a sum of characteristic polynomials.

Abstract

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.