Interlacement and Activities in Delta-Matroids

TL;DR

This paper extends the concept of activities to delta-matroids, providing new polynomial expansions that generalize classical graph and matroid invariants, including the Tutte and Bollobás-Riordan polynomials.

Contribution

It introduces an activities-based feasible-set expansion for a delta-matroid transition polynomial, generalizing existing graph and matroid polynomial expansions.

Findings

Feasible-set expansion for delta-matroid transition polynomial

Generalization of Tutte polynomial activities to delta-matroids

Derivation of expansions for Bollobás-Riordan and interlace polynomials

Abstract

We generalize theories of graph, matroid, and ribbon-graph activities to delta-matroids. As a result, we obtain an activities based feasible-set expansion for a transition polynomial of delta-matroids defined by Brijder and Hoogeboom. This result yields feasible-set expansions for the two-variable Bollob\'{a}s-Riordan and interlace polynomials of a delta-matroid. In the former case, the expansion obtained directly generalizes the activities expansions of the Tutte polynomial of graphs and matroids.