  In a recent paper, 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 that shrub-depth is monotone under taking vertex-minors, and that every
graph class of bounded shrub-depth can be obtained via vertex-minors of graphs
of bounded tree-depth. We also consider the same questions for bipartite graphs
and pivot-minors.
