Polynomial bounds for pathwidth

TL;DR

This paper proves that for hereditary graph classes, if pathwidth is bounded by a function of clique number, then it is actually bounded by a polynomial function, contrasting with previous results on treewidth.

Contribution

The paper establishes that the pathwidth analogue of a conjecture on treewidth holds true with polynomial bounds for hereditary graph classes.

Findings

Pathwidth is polynomially bounded by clique number in hereditary classes.

Contrasts with the failure of polynomial bounds for treewidth.

Provides a positive resolution for the pathwidth version of the conjecture.

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.