Compact Labelings For Efficient First-Order Model-Checking

We consider graph properties that can be checked from labels, i.e., bit sequences, of logarithmic length attached to vertices. We prove that there exists such a labeling for checking a first-order formula with free set variables in the graphs of every class that is emph{nicely locally cwd-decomposable}. This notion generalizes that of a emph{nicely locally tree-decomposable} class. The graphs of such classes can be covered by graphs of bounded emph{clique-width} with limited overlaps. We also consider such labelings for emph{bounded} first-order formulas on graph classes of emph{bounded expansion}. Some of these results are extended to counting queries.