Characterizing circle graphs with binomial partial Petrial polynomials

TL;DR

This paper characterizes connected circle graphs with binomial partial Petrial polynomials, proving that only paths among such graphs exhibit this property, thus solving an open problem in graph theory.

Contribution

It provides a complete characterization of connected circle graphs with binomial partial Petrial polynomials using local complementation, identifying paths as the unique case.

Findings

Paths are the only connected circle graphs with binomial partial Petrial polynomials.

The characterization uses local complementation techniques.

The result solves an open problem in the theory of circle graphs.

Abstract

The partial Petrial polynomial was first introduced by Gross, Mansour, and Tucker as a generating function that enumerates the Euler genera of all possible partial Petrials on a ribbon graph. Yan and Li later extended this polynomial invariant to circle graphs by utilizing the correspondence between circle graphs and bouquets. Their explicit computation demonstrated that paths produce binomial polynomials, specifically those containing exactly two non-zero terms. This discovery led them to pose a fundamental characterization problem: identify all connected circle graphs whose partial Petrial polynomial is binomial. In this paper, we solve this open problem in terms of local complementation and prove that for connected circle graphs, the binomial property holds precisely when the graph is a path.