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Partial-twuality polynomials of delta-matroids

Qi Yan, Xian'an Jin

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.

Partial-twuality polynomials of delta-matroids

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.
Paper Structure (12 sections, 16 theorems, 49 equations, 4 tables)

This paper contains 12 sections, 16 theorems, 49 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Set systems and widths
  4. Partial-twualities of set systems
  5. Partial-twualities of ribbon graphs
  6. Partial-twuality polynomials of ribbon graphs and set systems
  7. Binary and intersection graphs
  8. Primal and dual types
  9. Some properties of partial-twuality polynomials
  10. Partial-twuality for a single element
  11. Partial-twuality polynomials and intersection graphs
  12. Partial-twuality monomials

Key Result

Proposition 5

Let $D=(E, \mathcal{F})$ and $\widetilde{D}=(\widetilde{E}, \widetilde{\mathcal{F}})$ be set systems. Then for any $\bullet\in \mathcal{B}$,

Theorems & Definitions (34)

  • Definition 2: GMT2
  • Definition 3
  • Definition 4: CMNR
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Remark 7
  • Lemma 8: BRHH
  • Proposition 9
  • ...and 24 more