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arXiv · 1707.00359

Shrub-depth: Capturing Height of Dense Graphs

Robert Ganian·Petr Hliněný·Jaroslav Nešetřil·Jan Obdržálek·Patrice Ossona de Mendez
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Abstract

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.

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Robert Ganian, Petr Hliněný, Jaroslav Nešetřil, Jan Obdržálek, Patrice Ossona de Mendez. 2019-01-30. Shrub-depth: Capturing Height of Dense Graphs. https://doi.org/10.23638/lmcs-15(1%3A7)2019

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