Tight bound on treedepth in terms of pathwidth and longest path

TL;DR

This paper establishes a tight upper bound on the treedepth of graphs based on their pathwidth and longest path, showing the bound is nearly optimal.

Contribution

It provides a new, tight bound relating treedepth to pathwidth and longest path, improving understanding of graph structure.

Findings

Treedepth is at most 10ab for graphs with pathwidth less than a and no long paths.

The bound is tight up to a constant factor.

The result links treedepth, pathwidth, and path length in graphs.

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.