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Tight bound on treedepth in terms of pathwidth and longest path

Meike Hatzel, Gwenaël Joret, Piotr Micek, Marcin Pilipczuk, Torsten Ueckerdt, Bartosz Walczak

TL;DR

It is shown that every graph with pathwidth strictly less than a that contains no path on 2b is best possible up to a constant factor.

Abstract

We show that every graph with pathwidth strictly less than $a$ that contains no path on $2^b$ vertices as a subgraph has treedepth at most $10ab$. The bound is best possible up to a constant factor.

Tight bound on treedepth in terms of pathwidth and longest path

TL;DR

It is shown that every graph with pathwidth strictly less than a that contains no path on 2b is best possible up to a constant factor.

Abstract

We show that every graph with pathwidth strictly less than that contains no path on vertices as a subgraph has treedepth at most . The bound is best possible up to a constant factor.
Paper Structure (3 sections, 4 theorems, 21 equations, 1 figure)

This paper contains 3 sections, 4 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof

Key Result

Theorem 1

Every graph of treewidth less than $t$ with no complete binary tree of height $h$ as a minor and no $2^b$-vertex path has treedepth $\mathcal{O}(thb)$.

Figures (1)

  • Figure :

Theorems & Definitions (4)

  • Theorem 1: Czerwiński, Nadara, Pilipczuk CNP21
  • Theorem 2: Groenland, Joret, Nadara, Walczak GJNW23
  • Theorem 3
  • Theorem 4: Erde Erde18