Interlace Polynomials for Multimatroids and Delta-Matroids

TL;DR

This paper introduces a unified framework for interlace polynomials applicable to multimatroids and delta-matroids, revealing new relationships and simplifying proofs of known properties through combinatorial methods.

Contribution

It generalizes interlace polynomials to multimatroids and delta-matroids, providing new insights and more efficient proofs of their properties compared to graph-based approaches.

Findings

Unified framework for interlace polynomials on multimatroids and delta-matroids

New interrelationships between polynomials of multimatroids and graph polynomials

Proof of equivalence between tight 3-matroids and vf-safe delta-matroids

Abstract

We provide a unified framework in which the interlace polynomial and several related graph polynomials are defined more generally for multimatroids and delta-matroids. Using combinatorial properties of multimatroids rather than graph-theoretical arguments, we find that various known results about these polynomials, including their recursive relations, are both more efficiently and more generally obtained. In addition, we obtain several interrelationships and results for polynomials on multimatroids and delta-matroids that correspond to new interrelationships and results for the corresponding graphs polynomials. As a tool we prove the equivalence of tight 3-matroids and delta-matroids closed under the operations of twist and loop complementation, called vf-safe delta-matroids. This result is of independent interest and related to the equivalence between tight 2-matroids and even delta-matroids observed by Bouchet.