# Long Induced Paths in Minor-Closed Graph Classes and Beyond

### 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.