Multivariate Tutte polynomials of semimatroids

TL;DR

This paper extends classical graph and matroid polynomial invariants to semimatroids, establishing their fundamental properties and identities, thus broadening the theoretical framework of combinatorial invariants.

Contribution

It introduces multivariate Tutte and related polynomials for semimatroids, generalizing known invariants and proving key recurrence and convolution identities.

Findings

Established deletion-contraction recurrences for semimatroid polynomials.

Derived basis activities expansions and convolution identities.

Extended classical Tutte polynomial formulas to semimatroids.

Abstract

We introduce and investigate multivariate Tutte polynomials, dichromatic polynomials, subset-corank polynomials, size-corank polynomials, and rank generating polynomials of semimatroids, which generalize the corresponding polynomial invariants of graphs and matroids. We primarily establish their deletion-contraction recurrences, basis activities expansions, and various convolution identities. These findings naturally extend Kook-Reiner-Stanton's convolution formula and Kung's convolution-multiplication identities for the Tutte polynomials of graphs and matroids to semimatroids.