Treedepth and 2-treedepth in graphs with no long induced paths

TL;DR

This paper explores the relationship between treedepth and 2-treedepth in graphs without long induced paths, establishing bounds and answering open questions about their structural properties.

Contribution

It proves a bound on treedepth based on 2-treedepth and path length, and provides tight bounds for induced paths in graphs with bounded 2-treedepth or pathwidth.

Findings

Graphs with no long induced path and bounded 2-treedepth have bounded treedepth.

The function relating treedepth, 2-treedepth, and path length is determined up to a factor of 2.

Asymptotically tight bounds are established for long induced paths in graphs with bounded 2-treedepth or pathwidth.

Abstract

Huynh, Joret, Micek, Seweryn, and Wollan (Combinatorica, 2022) introduced a graph parameter, later referred to as 2-treedepth and denoted $\mathrm{td}_2(\cdot)$. The parameter is the natural 2-connected version of treedepth. For every graph, 2-treedepth is at most the treedepth but can be much smaller: long paths have arbitrary treedepth but 2-treedepth equal to 2. We prove a converse showing that every graph with no induced path on $t$ vertices and 2-treedepth at most $k$ has treedepth at most $g(k, t)$. In fact, we determine the value of the function $g$ up to a multiplicative factor of 2. Additionally, we give asymptotically tight bounds for the problem of forcing long induced paths in graphs with long paths and bounded 2-treedepth or bounded pathwidth. The latter result answers a question of Hilaire and Raymond (E-JC, 2024).