Interlace Polynomials for Multimatroids and 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.

35 pages, 3 figures

References in corpus (5)

• The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
• The Group Structure of Pivot and Loop Complementation on Graphs and Set Systems
• Nullity and Loop Complementation for Delta-Matroids
• Nullity Invariance for Pivot and the Interlace Polynomial
• Quaternary matroids are vf-safe

Cited by in corpus (15)

• On the interplay between embedded graphs and delta-matroids
• Matroids, Delta-matroids and Embedded Graphs
• Nullity and Loop Complementation for Delta-Matroids
• Binary matroids and local complementation
• Transition polynomial as a weight system for binary delta-matroids
• Quaternary Bicycle Matroids and the Penrose Polynomial for Delta-Matroids
• How many delta-matroids are there?
• The Universal Valuation of Coxeter Matroids
• Hopf algebras and Tutte polynomials
• Inductive tools for connected ribbon graphs, delta-matroids and multimatroids
• The adjacency matroid of a graph
• Delta-matroids whose twist polynomials are monomials
• Isotropic matroids II: Circle graphs
• Orienting Transversals and Transition Polynomials of Multimatroids
• The excluded 3-minors for vf-safe delta-matroids