On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic

We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is definable 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 quantification. Such quantifications are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this affects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, Various generating functions of MSOL definable graph properties. Finally, our results are also applicable to SAT and #SAT. (C) 2001 Elsevier Science B.V. All rights reserved.