Shrub-depth: Capturing Height of Dense Graphs

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...

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Vydané v:	Logical methods in computer science Ročník 15, Issue 1; číslo 1
Hlavní autori:	Ganian, Robert, Hliněný, Petr, Nešetřil, Jaroslav, Obdržálek, Jan, de Mendez, Patrice Ossona
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Logical Methods in Computer Science Association 01.01.2019 Logical Methods in Computer Science e.V
Predmet:	03b70, 05c75, 68r10 Combinatorics computer science - discrete mathematics computer science - logic in computer science Mathematics mathematics - combinatorics
ISSN:	1860-5974, 1860-5974
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Popis
Shrnutí:	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.
ISSN:	1860-5974 1860-5974
DOI:	10.23638/LMCS-15(1:7)2019