Tree-depth and vertex-minors

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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. (C) 2016 Elsevier Ltd. All rights reserved
Publisher
ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD
Issue Date
2016-08
Language
English
Article Type
Article
Keywords

CLIQUE-WIDTH; RANK-WIDTH; GRAPHS

Citation

EUROPEAN JOURNAL OF COMBINATORICS, v.56, pp.46 - 56

ISSN
0195-6698
DOI
10.1016/j.ejc.2016.03.001
URI
http://hdl.handle.net/10203/209711
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