Shrub-depth: Capturing Height of Dense Graphs - Ganian, Robert; Hliněný, Petr; Nešetřil, Jaroslav; Obdržálek, Jan; de Mendez, Patrice Ossona (Logical methods in computer science, 2019)
The recent increase of interest in the graph invariant called tree-depth and in its applications in algorithms and logic on graphs led to a natural question: is there an analogously useful "depth" notion also for dense graphs (say; one which is stable under graph complementation)? To this end, in a 2012 conference paper, a new notion of shrub-depth has been introduced, such that it is related to the established notion of clique-width in a similar way as tree-depth is related to tree-width. Since then shrub-depth has been successfully used in several research papers. Here we provide an in-depth review of the definition and basic properties of shrub-depth, and we focus on its logical aspects which turned out to be most useful. In particular, we use shrub-depth to give a characterization of the lower ${ω}$ levels of the MSO1 transduction hierarchy of simple graphs.