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 deenable in Monadic Second Order Logic. 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 quantiication. Such quantiications are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this aaects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, various generating functions of MSOL deenable graph properties. Finally, our results are also applicable to SAT and ]SAT.