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The recent increase of interest in the graph invariant called tree-depth andin its applications in algorithms and logic on graphs led to a naturalquestion: is there an analogously useful "depth" notion also for dense graphs(say; one which is stable under graph complementation)? To this end, in a 2012conference paper, a new notion of shrub-depth has been introduced, such that itis related to the established notion of clique-width in a similar way astree-depth is related to tree-width. Since then shrub-depth has beensuccessfully used in several research papers. Here we provide an in-depthreview of the definition and basic properties of shrub-depth, and we focus onits logical aspects which turned out to be most useful. In particular, we useshrub-depth to give a characterization of the lower ${\omega}$ levels of theMSO1 transduction hierarchy of simple graphs.

Shrub-depth: Capturing Height of Dense Graphs