On the interplay between embedded graphs and delta-matroids

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 deltamatroids. 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.

Delta-matroids as subsystems of sequences of Higgs lifts

“…Note that S/e = (S * e)\e and (S * A) * A = S. The dual S * of S is S * E. Note that twists of even set systems are even. However, apart from the dual, the twists of a matroid E(M ), B(M ) are generally not matroids, as discussed in [15,Theorem 3.4].…”

“…(SE) for all triples (X, Y, u) with X and Y in F and u ∈ X Y , there is a v ∈ X Y (perhaps u itself) such that X {u, v} is in F. Just as there is a mutually-enriching interplay between matroid theory and graph theory, the theory of delta-matroids has substantial connections with the theory of embedded graphs; see [14,15].…”

The excluded 3-minors for vf-safe delta-matroids

“…Just as there is a mutually-enriching interplay between matroid theory and graph theory, the theory of delta-matroids has substantial connections with the theory of embedded graphs or equivalently ribbon graphs; see [17,18]. Brijder and Hoogeboom [12,13,14] introduced the operation of loop complementation, which we define in the next paragraph.…”

“…There is a natural operation of embedded graphs, namely partial Petriality, to which loop complementation corresponds. More precisely if two embedded graphs are partial Petrials of each other then their ribbon graphic delta-matroids are related by a loop complementation [18,Section 4]. For a delta-matroid D and element e ∈ E(D), the set system D + e need not be a deltamatroid.…”

“…Note that (S * A) * A = S. The dual S * of S is S * E. In contrast with loop complementation, each twist of a delta-matroid is a delta-matroid. Apart from the dual, the twists of a matroid (E(M ), B(M )) are generally not matroids, as discussed in [18,Theorem 3.4].…”