# Branch-depth: Generalizing tree-depth of graphs

## TL;DR

This paper introduces the branch-depth concept for connectivity functions, generalizing tree-depth of graphs, and establishes its relation to shrub-depth and matroid properties, providing a unified framework for graph and matroid complexity measures.

## Contribution

It defines branch-depth for connectivity functions, linking it to existing graph parameters and extending the concept to matroids, with key theorems connecting boundedness of these parameters.

## Key findings

- Bounded tree-depth classes correspond to bounded branch-depth of connectivity functions.
- Bounded shrub-depth classes correspond to bounded branch-depth of rank functions.
- Matroids over fixed finite fields with no large circuits are well-quasi-ordered.

## Abstract

We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A $ of $E$ we let $\lambda_G (A)$ be the number of vertices incident with an edge in $A$ and an edge in $E \setminus A$. For a subset $X$ of $V$, let $\rho_G(X)$ be the rank of the adjacency matrix between $X$ and $V \setminus X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions $\lambda_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions $\rho_G$ has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.