Irreducibility of the Tutte polynomial of an embedded graph

TL;DR

This paper characterizes when the ribbon graph polynomial of an embedded graph is irreducible, showing it depends on the graph's structure, specifically whether it is a disjoint union or join of embedded graphs.

Contribution

It establishes a necessary and sufficient condition for the irreducibility of the ribbon graph polynomial based on the graph's decomposition properties.

Findings

Ribbon graph polynomial is irreducible iff the graph is not a disjoint union or join.

The result parallels the classical irreducibility condition for the Tutte polynomial.

Provides a structural criterion for polynomial irreducibility in embedded graphs.

Abstract

We prove that the ribbon graph polynomial of a graph embedded in an orientable surface is irreducible if and only if the embedded graph is neither the disjoint union nor the join of embedded graphs. This result is analogous to the fact that the Tutte polynomial of a graph is irreducible if and only if the graph is connected and non-separable.