Polynomial bounds for pathwidth

Abstract

Dallard, Milani\v{c}, and \v{S}torgel conjectured that for a hereditary graph class $\mathcal{G}$, if there is some function $f:\mathbb{N}\to\mathbb{N}$ such that every graph $G\in \mathcal{G}$ with clique number $\omega(G)$ has treewidth at most $f(\omega(G))$, then there is a polynomial function $f$ with the same property. Chudnovsky and Trotignon refuted this conjecture in a strong sense, showing that neither polynomial nor any prescribed growth can be guaranteed in general. Here we prove that, in stark contrast, the analog of the Dallard-Milani\v{c}-\v{S}torgel conjecture for pathwidth is true: For every hereditary graph class $\mathcal{G}$, if the pathwidth of every graph in $\mathcal{G}$ is bounded by some function of its clique number, then the pathwidth of every graph in $\mathcal{G}$ is bounded by a polynomial function of its clique number.