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arXiv · 1911.00230

Obstructions for bounded shrub-depth and rank-depth

O-joung Kwon·Rose McCarty·Sang-il Oum·Paul Wollan
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Abstract

Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph. It is well known that a graph has large tree-depth if and only if it has a long path as a subgraph. We prove an analogous statement for shrub-depth and rank-depth, which was conjectured by Hlin\v{e}n\'y, Kwon, Obdr\v{z}\'alek, and Ordyniak [Tree-depth and vertex-minors, European J.~Combin. 2016]. Namely, we prove that a graph has large rank-depth if and only if it has a vertex-minor isomorphic to a long path. This implies that for every integer $t$, the class of graphs with no vertex-minor isomorphic to the path on $t$ vertices has bounded shrub-depth.

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O-joung Kwon, Rose McCarty, Sang-il Oum, Paul Wollan. 2019-11-01. Obstructions for bounded shrub-depth and rank-depth. https://doi.org/10.1016/j.jctb.2021.01.005

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