Characterize connected graphs with binomial partial Petrial polynomials

Determine whether every connected graph whose partial Petrial polynomial is a binomial must be a path; if not, characterize all connected graphs whose partial Petrial polynomials are binomials.

Background

The paper defines the partial Petrial polynomial of a circle graph by enumerating partial Petrials according to Euler genus. It proves that complete graphs have nonzero coefficients in every degree from 1 through the number of vertices, while Theorem 2 shows that the partial Petrial polynomial of every path is a binomial, with explicit coefficients depending on the parity of the number of vertices.

These results motivate the unresolved converse question: whether the binomial form uniquely characterizes paths among connected graphs. The authors explicitly leave open both the necessity of being a path and, if that necessity fails, the broader classification of connected graphs with binomial partial Petrial polynomials.

References

This raises the question of whether the converse holds: If the partial Petrial polynomial of a connected graph $G$ is a binomial, must $G$ necessarily be a path? If not, can we characterize the graphs whose partial Petrial polynomials are binomials?

— Partial Petrial polynomials for complete graphs and paths  (2501.04186 - Yan et al., 7 Jan 2025) in Section 4, Concluding remarks, immediately following Theorem 2