Abstract. We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce two classes of (hyper)graph polynomials definable in second order logic, and outline a research program for their classification in terms of definability and complexity considerations, and various notions of reducibilities. 1

We consider the two-variable interlace polynomial introduced by Arratia, Bollobás and Sorkin (2004). We develop two graph transformations which allow us to derive point-to-point reductions for the interlace polynomial. Exploiting these reductions we obtain new results concerning the computational complexity of evaluating the interlace polynomial at a fixed point. Regarding exact evaluation, we prove that the interlace polynomial is #P-hard to evaluate at every point of the plane, except at one line, where it is trivially polynomial time computable, and four lines and two points, where the complexity mostly is still open. This solves a problem posed by Arratia, Bollobás and Sorkin (2004). In particular, we observe that three specializations of the two-variable interlace polynomial, the vertex-nullity interlace polynomial, the vertexrank interlace polynomial and the independent set polynomial, are almost everywhere #P-hard to evaluate, too. For the independent set polynomial, our reductions allow us to prove that it is even hard to approximate at every point except at −1 and 0.