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 nicely locally clique‐width‐decomposable. This notion generalizes that of a nicely locally tree‐decomposable class. The graphs of such classes can be covered by graphs of bounded clique‐width with limited overlaps. We also consider such labelings for bounded first‐order formulas on graph classes of bounded expansion. Some of these results are extended to counting queries.