The monadic second-order logic of graphs. I. recognizable sets of finite graphs
Information and Computation
A Tutte polynomial for signed graphs
Discrete Applied Mathematics - Combinatorics and complexity
Easy problems for tree-decomposable graphs
Journal of Algorithms
The monadic second-order logic of graphs VII: graphs as relational structures
Theoretical Computer Science - Special issue on logic and applications to computer science
Tutte polynomials computable in polynomial time
Discrete Mathematics - Algebraic graph theory; a volume dedicated to Gert Sabidussi
Complexity: knots, colourings and counting
Complexity: knots, colourings and counting
Monotone monadic SNP and constraint satisfaction
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Splitting formulas for Tutte polynomials
Journal of Combinatorial Theory Series B
An algorithm for the Tutte polynomials of graphs of bounded treewidth
Discrete Mathematics
Upper bounds to the clique width of graphs
Discrete Applied Mathematics
Colored Tutte polynomials and Kaufman brackets for graphs of bounded tree width
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Discrete Applied Mathematics - Special issue on international workshop of graph-theoretic concepts in computer science WG'98 conference selected papers
Journal of Combinatorial Theory Series B
Reduction algorithms for graphs of small treewidth
Information and Computation
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The Complexity of First-Order and Monadic Second-Order Logic Revisited
LICS '02 Proceedings of the 17th Annual IEEE Symposium on Logic in Computer Science
Treewidth: Algorithmoc Techniques and Results
MFCS '97 Proceedings of the 22nd International Symposium on Mathematical Foundations of Computer Science
Invariant Definability (Extended Abstract)
KGC '97 Proceedings of the 5th Kurt Gödel Colloquium on Computational Logic and Proof Theory
Farrell polynomials on graphs of bounded tree width
Advances in Applied Mathematics - Special issue on: Formal power series and algebraic combinatorics in memory of Rodica Simion, 1955-2000
Back and Forth between Guarded and Modal Logics
LICS '00 Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science
The parametrized complexity of knot polynomials
Journal of Computer and System Sciences - Special issue on Parameterized computation and complexity
Fusion in relational structures and the verification of monadic second-order properties
Mathematical Structures in Computer Science
A Tutte Polynomial for Coloured Graphs
Combinatorics, Probability and Computing
Evaluating the Tutte Polynomial for Graphs of Bounded Tree-Width
Combinatorics, Probability and Computing
Parameterized Complexity
Farrell polynomials on graphs of bounded tree width
Advances in Applied Mathematics - Special issue on: Formal power series and algebraic combinatorics in memory of Rodica Simion, 1955-2000
The Tutte Polynomial for Matroids of Bounded Branch-Width
Combinatorics, Probability and Computing
On the colored Tutte polynomial of a graph of bounded treewidth
Discrete Applied Mathematics
Counting truth assignments of formulas of bounded tree-width or clique-width
Discrete Applied Mathematics
An extension of the bivariate chromatic polynomial
European Journal of Combinatorics
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Tutte polynomials are important graph invariants with rich applications in combinatorics, topology, knot theory, coding theory and even physics. The Tutte polynomial T(G, X, Y) is a polynomial in Z[X, Y] which depends on a graph G. Computing the coefficients of T(G, X, Y), and even evaluating T(G, X, Y) at specific points (x, y) is #P hard by a result of Jaeger et al. (Math. Proc. Cambridge Philos. Soc. 108 (1989) 35). On the other hand, Andrzejak (Discrete Math. 190 (1998) 39-54) and Noble (Combin. Probab. Comput. 7 (1998) 307-321) have shown independently, that, if G is a graph of bounded tree width, computing T(G, X, Y) can be done in polynomial time. We extend this result to the signed Tutte polynomials introduced in 1989 by Kauffman and the colored Tutte polynomials introduced in 1999 by Bollobas and Riordan. This allows us to prove similar results for the Jones polynomials and Kauffman brackets for knots and links which have a signed graph presentation of bounded tree width. Jones polynomials and Kauffman polynomials are the most prominent invariants of knot theory. For alternating links, they are easily computable from the Tutte polynomials of the signed graph representing the link by a result of Thistlethwaite (1988). For general links one has to use the colored Tutte polynomial instead. Knots and links can be presented as labeled planar graphs. The tree width of a link L is defined as the tree width of its graphical presentation D(L) as crossing diagrams. We show that for (not necessarily alternating) knots and links of tree width at most k, even the Kauffman square bracket [L] introduced by Bollobas and Riordan can be computed in polynomial time. Hence, the classical Kauffman bracket (L) and the Jones polynomial of links of tree width at most k are computable in polynomial time. Our proof is based on, but extends considerably previous work by B. Courcelle, U. Rotics and the author. It also gives a new proof of the result for Tutte polynomials and generalizes to a wide class of polynomials defined as generating functions definable in Monadic Second Order Logic with order, but invariant under it.