On the xed parameter complexity of graph enumeration problems de nable in monadic second-order logic

We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is de nable in monadic second-order logic (MSOL). We show that for bounded tree-width these problems are solvable in polynomial time. The same holds for bounded clique width in the cases, where the decomposition, which establishes the bound on the clique-width, can be computed in polynomial time and for problems expressible by monadic second-order formulas without edge set quanti cation. Such quanti cations are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this a ects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, various generating functions of MSOL de nable graph properties. Finally, our results are also applicable to SAT and ]SAT . ? 2001 Elsevier Science B.V. All rights reserved.

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