Obstructions for bounded shrubdepth and rankdepth
Abstract
Shrubdepth and rankdepth are dense analogues of the treedepth of a graph. It is well known that a graph has large treedepth if and only if it has a long path as a subgraph. We prove an analogous statement for shrubdepth and rankdepth, which was conjectured by Hliněný, Kwon, Obdržálek, and Ordyniak [Treedepth and vertexminors, European J.~Combin. 2016]. Namely, we prove that a graph has large rankdepth if and only if it has a vertexminor isomorphic to a long path. This implies that for every integer $t$, the class of graphs with no vertexminor isomorphic to the path on $t$ vertices has bounded shrubdepth.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.00230
 Bibcode:
 2019arXiv191100230K
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics
 EPrint:
 19 pages, 5 figures