# Graphs of bounded depth-$2$ rank-brittleness

## TL;DR

This paper characterizes certain graph classes closed under vertex-minors, using a tree-based measure called depth-2 rank-brittleness, and shows large values imply the presence of specific subgraphs as vertex-minors.

## Contribution

It introduces the concept of depth-2 rank-brittleness and provides a characterization linking it to forbidden subgraphs and vertex-minors.

## Key findings

- Graphs with large depth-2 rank-brittleness contain specific subgraphs as vertex-minors.
- The characterization involves a tree of radius 2 with leaves labeled by graph vertices.
- The measure of width is based on maximum cut-rank induced by splitting internal nodes.

## Abstract

We characterize classes of graphs closed under taking vertex-minors and having no $P_n$ and no disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ for some $n$. Our characterization is described in terms of a tree of radius $2$ whose leaves are labelled by the vertices of a graph $G$, and the width is measured by the maximum possible cut-rank of a partition of $V(G)$ induced by splitting an internal node of the tree to make two components. The minimum width possible is called the depth-$2$ rank-brittleness of $G$. We prove that for all $n$, every graph with sufficiently large depth-$2$ rank-brittleness contains $P_n$ or disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ as a vertex-minor.