Summary: It has been proved in Frick and Grohe that properties of graphs or other relational structures that are definable in first-order logic can be decided in linear time when the input structures are restricted to come from a locally tree-decomposable class of structures. Important examples of such classes are the class of planar graphs and classes of graphs of bounded degree.
In this paper we consider more general computational problems than decision problems, which are induced by formulas with free variables. In this generalized setting we investigate the construction (find a satisfying assignment), listing (all satisfying assignments) and counting (compute the number of satisfying assignments) problems for formulas of first-order logic.
We show that each of these problems can be solved in linear time on locally tree-decomposable classes of structures. For instance, we devise an algorithm that, given a planar graph \(\mathcal G\) and a first-order logic formula \(\phi(\bar x)\), computes a \(\bar a \in G\) such that \(\mathcal G \models \phi(\bar a)\) in time \(f(||\phi||) \cdot ||\mathcal G||\) for some computable function \(f\) (the construction case). Accordingly, we obtain linear time algorithms for the counting and listing cases.