Long induced paths in minor-closed graph classes and beyond

TL;DR

This paper establishes new bounds on the size of induced paths in graphs with bounded pathwidth and treewidth, extending to broader classes, with significant exponential and logarithmic improvements over previous results.

Contribution

It introduces exponential and logarithmic bounds for induced paths in graphs of bounded pathwidth and treewidth, generalizing prior polylogarithmic bounds for interval graphs.

Findings

Graphs with pathwidth less than k and a path of length n have an induced path of length at least (1/3) n^{1/k}.

Graphs with treewidth less than k and a path of length n contain an induced path of at least (1/4) (log n)^{1/k}.

For classes closed under topological minors, large paths imply the existence of substantial induced paths, with bounds involving powers of logarithms.

Abstract

In this paper we show that every graph of pathwidth less than $k$ that has a path of order $n$ also has an induced path of order at least $\frac{1}{3} n^{1/k}$. This is an exponential improvement and a generalization of the polylogarithmic bounds obtained by Esperet, Lemoine and Maffray (2016) for interval graphs of bounded clique number. We complement this result with an upper-bound. This result is then used to prove the two following generalizations: - every graph of treewidth less than $k$ that has a path of order $n$ contains an induced path of order at least $\frac{1}{4} (\log n)^{1/k}$; - for every non-trivial graph class that is closed under topological minors there is a constant $d \in (0,1)$ such that every graph from this class that has a path of order $n$ contains an induced path of order at least $(\log n)^d$. We also describe consequences of these results beyond graph classes that are closed under topological minors.