Summary: Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Three 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.