Partial-twuality polynomials of delta-matroids

TL;DR

This paper extends the concept of partial-twuality polynomials from ribbon graphs to delta-matroids, exploring their properties, implications, and characterizations in various delta-matroid classes.

Contribution

It introduces and studies analogues of partial-twuality polynomials for delta-matroids, including their properties and how intersection graphs determine these polynomials.

Findings

Intersection graphs determine partial-twuality polynomials for bouquets and normal binary delta-matroids.

Characterization of vf-safe delta-matroids with single-term partial-twuality polynomials.

Various properties of partial-twuality polynomials of set systems are analyzed.

Abstract

Gross, Mansour and Tucker introduced the partial-twuality polynomial of a ribbon graph. Chumutov and Vignes-Tourneret posed a problem: it would be interesting to know whether the partial duality polynomial and the related conjectures would make sense for general delta-matroids. In this paper we consider analogues of partial-twuality polynomials for delta-matroids. Various possible properties of partial-twuality polynomials of set systems are studied. We discuss the numerical implications of partial-twualities on a single element and prove that the intersection graphs can determine the partial-twuality polynomials of bouquets and normal binary delta-matroids, respectively. Finally, we give a characterization of vf-safe delta-matroids whose partial-twuality polynomials have only one term.