Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decomposi-tions are the best known ones. On graphs of tree-width at most k, i.e., that have tree decompositions of width at most k, where k is fixed, every decision or optimization problem expressible in monadic second-order logic has a linear algorithm. We prove that this is also the case for graphs of clique-width at most k, where this complexity measure is associated with hierarchical decompositions of another type, and where logical formulas are no longer allowed to use edge set quantifications. We develop applications to several classes of graphs that include cographs and are, like cographs, defined by forbidding subgraphs with "too many" induced paths with four vertices.

Abstract. Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones. On graphs of tree-width at most k, i.e., that have tree decompositions of width at most k, where k is fixed, every decision or optimization problem expressible in monadic second-order logic has a linear algorithm. We prove that this is also the case for graphs of clique-width at most k, where this complexity measure is associated with hierarchical decompositions of another type, and where logical formulas are no longer allowed to use edge set quantifications. We develop applications to several classes of graphs that include cographs and are, like cographs, defined by forbidding subgraphs with "too many" induced paths with four vertices.

The classical Feferman-Vaught Theorem for First Order Logic explains how to compute the truth value of a ÿrst order sentence in a generalized product of ÿrst order structures by reducing this computation to the computation of truth values of other ÿrst order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by L auchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and deÿnability theory.Here we give a uniÿed presentation, including some new results, of how to use the FefermanVaught Theorem, and some new variations thereof, algorithmically in the case of Monadic Second Order Logic MSOL.We then extend the technique to graph polynomials where the range of the summation of the monomials is deÿnable in MSOL. Here the Feferman-Vaught Theorem for these polynomials generalizes well known splitting theorems for graph polynomials. Again, these can be used algorithmically.Finally, we discuss extensions of MSOL for which the Feferman-Vaught Theorem holds as well. (J.A. Makowsky).1 At the occasion of his 75th birthday. It is also a tribute to Feferman's work in model theory, which went seemingly unnoticed at the celebrations of his 70th birthday. Feferman returned several times to topics related to the preservation of truth under various model theoretic constructions, so in [54][55][56][57][58]60]. The FefermanVaught Theorem can also be viewed as falling into this category.

We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is deÿnable 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 quantiÿcation. Such quantiÿcations 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 deÿnable graph properties. Finally, our results are also applicable to SAT and ]SAT .

We study algorithms for SAT and its generalized version GENSAT, the problem of computing the number of satisfying assignments of a set of propositional clauses . For this purpose we consider the clauses given by their incidence graph, a signed bipartite graph SI( ), and its derived graphs I ( ) and P ( ).It is well known, that, given a graph of tree-width k, a k-tree decomposition can be found in polynomial time. Very recently Oum and Seymour have shown that, given a graph of clique-width k, a (2 3k+2 − 1)-parse tree witnessing clique-width can be found in polynomial time.In this paper we present an algorithm for GENSAT for formulas of bounded tree-width k which runs in time 4 k (n + n 2 · log 2 (n)), where n is the size of the input. The main ingredient of the algorithm is a splitting formula for the number of satisfying assignments for a set of clauses where the incidence graph I ( ) is a union of two graphs G 1 and G 2 with a shared induced subgraph H of size at most k. We also present analogue improvements for algorithms for formulas of bounded clique-width which are given together with their derivation.This considerably improves results for SAT, and hence also for SAT, previously obtained by Courcelle et al. [On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic, Discrete Appl. Math. 108 (1-2) (2001) 23-52].