Complexity of the Bollobás-Riordan Polynomial

The coloured Tutte polynomial by Bollobás and Riordan is, as a generalisation of the Tutte polynomial, the most general graph polynomial for coloured graphs that satisfies certain contractiondeletion identities. Jaeger, Vertigan, and Welsh showed that the classical Tutte polynomial is #P-hard to evaluate almost everywhere by establishing reductions along curves and lines. We establish a similar result for the coloured Tutte polynomial on integral domains. To establish this result in a proper way, we introduce a new kind of reductions, uniform algebraic reductions, that are well-suited to investigate the complexity of the evaluation of graph polynomials. Our main result identifies a small, algebraic set of exceptional points, with the property that there exists a uniform algebraic reduction that reduces the evaluation problem for the coloured Tutte polynomial at any one non-exceptional point to any other non-exceptional point. On the way we also obtain a stronger result and a new proof for the difficult evaluations in the complexity analysis for the classical Tutte polynomial.