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P. Hlinený, O. Kwon, J. Obdrzalek, S. Ordyniak:
"Tree-depth and vertex-minors";
European Journal of Combinatorics, 56 (2016), 46 - 56.

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.