Tree-depth and vertex-minors. (August 2016)

• Record Type: Journal Article
• Authors: Hliněný, Petr; Kwon, O-joung; Obdržálek, Jan; Ordyniak, Sebastian
• Is Part Of: European journal of combinatorics. Volume 56(2016:Aug.)
• Page Start: 46
• Page End: 56
• Publication Date: 2016-08
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2016.03.001
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: In a recent paper Kwon and Oum (2014), Kwon and Oum claim that every graph of bounded rank-width is a pivot-minor of a graph of bounded tree-width (while the converse has been known true already before). We study the analogous questions for "depth" parameters of graphs, namely for the tree-depth and related new shrub-depth. We show how a suitable adaptation of known results implies that shrub-depth is monotone under taking vertex-minors, and we prove that every graph class of bounded shrub-depth can be obtained via vertex-minors of graphs of bounded tree-depth. While we exhibit an example that pivot-minors are generally not sufficient (unlike Kwon and Oum (2014)) in the latter statement, we then prove that the bipartite graphs in every class of bounded shrub-depth can be obtained as pivot-minors of graphs of bounded tree-depth.