## Ganian, Robert and Hliněný, Petr and Nešetřil, Jaroslav and Obdržálek, Jan and de Mendez, Patrice Ossona - Shrub-depth: Capturing Height of Dense Graphs

lmcs:3798 - Logical Methods in Computer Science, January 31, 2019, Volume 15, Issue 1
Shrub-depth: Capturing Height of Dense Graphs

Authors: Ganian, Robert and Hliněný, Petr and Nešetřil, Jaroslav and Obdržálek, Jan and de Mendez, Patrice Ossona

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 ${\omega}$ levels of the MSO1 transduction hierarchy of simple graphs.

Source : oai:arXiv.org:1707.00359
DOI : 10.23638/LMCS-15(1:7)2019
Volume: Volume 15, Issue 1
Published on: January 31, 2019
Submitted on: July 19, 2017
Keywords: Computer Science - Logic in Computer Science,Computer Science - Discrete Mathematics,Mathematics - Combinatorics,03B70, 05C75, 68R10