Evaluating the Tutte Polynomial for Graphs of Bounded Tree-Width

Abstract

It is known that evaluating the Tutte polynomial, T (G; x, y), of a graph, G, is #P-hard at all but eight specific points and one specific curve of the (x, y)-plane. In contrast we show that if k is a fixed constant then for graphs of tree-width at most k there is an algorithm that will evaluate the polynomial at any point using only a linear number of multiplications and additions.

Computing Graph Polynomials on Graphs of Bounded Clique-Width

“…Similar results for polygraphs, which are special case of graph classes of bounded path-width, were obtained already in [BGMP86]. For the Tutte polynomial, and hence for the chromatic polynomial, similar bounds can be obtained from [And98,Nob98].…”

“…This was extended to arbitrary fixed tree-width k independently by A. Andrzejak [And98] and S. Noble [Nob98], and therefore also holds for the chromatic polynomial. Actually, they showed that computing the Tutte polynomial is in FPT on graph classes of tree-width at most k with computation time roughly f (k)n 3 where f (k) is simply exponential in k and the tree-decomposition of the graph is given in advance.…”

Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth

“…These similarities suggest that evaluating the interlace polynomial using tree decompositions might work completely analogously to the respective approaches for the Tutte polynomial [And98,Nob98]. This is not the case because of the following problems.…”

“…The interlace polynomial, in contrast, needs more and more complicated recursions when generalizing the vertex-nullity interlace polynomial to the multivariate interlace polynomial 1 (see [Cou08,Proposition 12]). When we consider Noble's algorithm [Nob98] and concentrate on the definition of the Tutte/interlace polynomial by sums involving ranks, another problem emerges. In the Tutte case, the rank is an easy to understand graph theoretic value, namely the number of vertices minus the number of connected components.…”

“…Our approach is dynamic programming similar to the work of Noble [Nob98]. Let G be a graph for which we want to evaluate the interlace polynomial and ({X i }, T ) a nice tree decomposition of G. For each node i of the tree decomposition, we have defined the graph G i that consists of all vertices in the bags below i that are not in X i .…”