Quasi-tree expansion for the Bollob\'as-Riordan-Tutte polynomial

TL;DR

This paper introduces a quasi-tree expansion for the Bollobás-Riordan-Tutte polynomial, extending the spanning tree expansion of the Tutte polynomial to oriented ribbon graphs with a new combinatorial approach.

Contribution

It generalizes the spanning tree expansion of the Tutte polynomial to a quasi-tree expansion for the Bollobás-Riordan-Tutte polynomial in oriented ribbon graphs.

Findings

Provides a new combinatorial expansion for the polynomial.

Extends classical Tutte polynomial techniques to ribbon graphs.

Enables new computations for embedded graph invariants.

Abstract

Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. The Bollob\'as-Riordan-Tutte polynomial is a three-variable polynomial that extends the Tutte polynomial to oriented ribbon graphs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We generalize the spanning tree expansion of the Tutte polynomial to a quasi-tree expansion of the Bollob\'as-Riordan-Tutte polynomial.