On the constant partial-dual polynomials of hypermaps

TL;DR

This paper introduces the partial-dual polynomial for hypermaps, explores its properties, and characterizes when it is constant, especially for prime connected hypermaps, linking it to planarity and hyperedge count.

Contribution

It extends the concept of partial-dual polynomials from ribbon graphs to hypermaps and characterizes their behavior for specific hypermap classes.

Findings

Partial-dual polynomial defined for hypermaps.

Characterization of hypermaps with non-zero constant term.

Constant partial-dual polynomial iff hypermap is plane with one hyperedge.

Abstract

In this paper, we introduce the partial-dual polynomial for hypermaps, extending the concept from ribbon graphs. We discuss the basic properties of this polynomial and characterize it for hypermaps with exactly one hypervertex containing a non-zero constant term. Additionally, we show that the partial-dual polynomial of a prime connected hypermap $H$ is constant if and only if $H$ is a plane hypermap with a single hyperedge.