# Improved bounds for the excluded-minor approximation of treedepth

## TL;DR

This paper establishes improved bounds relating treedepth and treewidth, providing new structural insights and algorithms for approximating treedepth in graphs, with implications for sparse graph theory.

## Contribution

It introduces tighter bounds connecting treedepth and treewidth, and offers polynomial-time algorithms for treedepth approximation with better guarantees than previous results.

## Key findings

- Bound of treedepth $ o$ either large treewidth or specific subgraphs
- Improved bound from $ ilde{ ext{Omega}}(k^5)$ to $ ext{Omega}(k^3)$ for treedepth
- Polynomial-time algorithm for treedepth approximation with improved factor

## Abstract

Treedepth, a more restrictive graph width parameter than treewidth and pathwidth, plays a major role in the theory of sparse graph classes. We show that there exists a constant $C$ such that for every positive integers $a,b$ and a graph $G$, if the treedepth of $G$ is at least $Cab$, then the treewidth of $G$ is at least $a$ or $G$ contains a subcubic (i.e., of maximum degree at most $3$) tree of treedepth at least $b$ as a subgraph. As a direct corollary, we obtain that every graph of treedepth $\Omega(k^3)$ is either of treewidth at least $k$, contains a subdivision of full binary tree of depth $k$, or contains a path of length $2^k$. This improves the bound of $\Omega(k^5 \log^2 k)$ of Kawarabayashi and Rossman [SODA 2018]. We also show an application of our techniques for approximation algorithms of treedepth: given a graph $G$ of treedepth $k$ and treewidth $t$, one can in polynomial time compute a treedepth decomposition of $G$ of width $\mathcal{O}(kt \log^{3/2} t)$. This improves upon a bound of $\mathcal{O}(kt^2 \log t)$ stemming from a tradeoff between known results. The main technical ingredient in our result is a proof that every tree of treedepth $d$ contains a subcubic subtree of treedepth at least $d \cdot \log_3 ((1+\sqrt{5})/2)$.