Tensor product formulas for the Bollob\'as-Riordan and Krushkal polynomials

Abstract

Brylawski's tensor product formula expresses the Tutte polynomial of the tensor product of two graphs in terms of Tutte polynomials arising from the tensor factors. Analogous tensor product formulas are known for the ribbon graph polynomial and transition polynomials of graphs embedded in surfaces, as well as for the Bollob\'as-Riordan polynomial in some special cases. We define the tensor product of graphs embedded in pseudo-surfaces and use this to generalize and unify all of the above results, providing Brylawski-style formulas for both the Bollob\'as-Riordan and Krushkal polynomials.