Unavoidable pivot-minors in graphs of large rank-depth

TL;DR

The paper proves that graphs with large rank-depth necessarily contain specific complex pivot-minors, resolving an open problem and advancing understanding of dense graph parameters related to tree-depth.

Contribution

It establishes that large rank-depth graphs always contain certain pivot-minors, specifically paths or bipartite-like structures, answering a question posed in 2021.

Findings

Large rank-depth graphs contain specific pivot-minors.

Graphs of large rank-depth include paths or bipartite-like structures as pivot-minors.

The result connects dense graph parameters with minor-closed properties.

Abstract

Shrub-depth and rank-depth are related graph parameters that are dense analogs of tree-depth. We prove that for every positive integer $t$, every graph of sufficiently large rank-depth contains a pivot-minor isomorphic to a path on $t$ vertices or a graph consisting of two disjoint cliques of size $t$ joined by a half graph. This answers an open problem raised by Kwon, McCarty, Oum, and Wollan in 2021.