Obstructions for bounded shrub-depth and rank-depth

TL;DR

This paper establishes a fundamental connection between large rank-depth in graphs and the presence of long paths as vertex-minors, extending the understanding of graph parameters related to shrub-depth and tree-depth.

Contribution

It proves that large rank-depth in a graph is characterized by the existence of a long path as a vertex-minor, confirming a conjecture and linking graph depth parameters.

Findings

Graphs with large rank-depth contain long paths as vertex-minors.

Classes of graphs excluding long path vertex-minors have bounded shrub-depth.

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.