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Complexity of the Bollobás–Riordan Polynomial. Exceptional Points and Uniform Reductions

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Abstract

The coloured Tutte polynomial by Bollobás and Riordan is, as a generalization of the Tutte polynomial, the most general graph polynomial for coloured graphs that satisfies certain contraction-deletion 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 capture the algebraic flavour and the uniformity inherent in this type of result, we introduce a new kind of reductions, uniform algebraic reductions, that are well-suited to investigate the evaluation complexity of graph polynomials. Our main result identifies a small, algebraic set of exceptional points and says that the evaluation problem of the coloured Tutte is equivalent for all non-exceptional points, under polynomial-time uniform algebraic reductions. On the way we obtain a self-contained proof for the difficult evaluations of the classical Tutte polynomial.

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Correspondence to Holger Dell.

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H. Dell was supported by the Deutsche Forschungsgemeinschaft within the research training group “Methods for Discrete Structures” (GRK 1408).

J.A. Makowsky was partially supported by a grant of the Israel Science Foundation (2007–2010) and the Fund for the Promotion of Research of the Technion—Israel Institute of Technology.

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Bläser, M., Dell, H. & Makowsky, J.A. Complexity of the Bollobás–Riordan Polynomial. Exceptional Points and Uniform Reductions. Theory Comput Syst 46, 690–706 (2010). https://doi.org/10.1007/s00224-009-9213-7

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  • DOI: https://doi.org/10.1007/s00224-009-9213-7

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