The Tutte Polynomial for Matroids of Bounded Branch-Width

It is a classical result of Jaeger, Vertigan and Welsh that evaluating the Tutte polynomial of a graph is #P -hard in all but few special points. On the other hand, several papers in past years have shown that the Tutte polynomial of a graph can be efficiently computed for graphs of bounded tree-width. In this paper we present a recursive formula computing the Tutte polynomial of a matroid M represented over a finite field (which includes all graphic matroids), using a so called parse tree of a branch-decomposition of M . This formula provides an algorithm computing the Tutte polynomial for a representable matroid of bounded branch-width in polynomial time with a fixed exponent.