Induced subgraphs and tree decompositions XIX. Thetas and forests

TL;DR

This paper establishes conditions under which the treewidth of graphs in a hereditary class is polynomially bounded by their clique number, focusing on theta-free graphs excluding certain line graphs of subdivisions of walls.

Contribution

It proves that if a hereditary class excludes line graphs of subdivisions of walls and contains only forests, then the treewidth is polynomially bounded by the clique number.

Findings

Treewidth is polynomially bounded by clique number under specified conditions.

Both conditions of excluding line graphs of subdivisions of walls and being a forest are necessary.

The result is optimal, with both conditions being essential.

Abstract

Let $H$ be a graph and let $\mathcal{C}$ be a hereditary class of theta-free graphs such that $H\notin \mathcal{C}$. We prove that if (a) $H$ is a forest; and (b) $\mathcal{C}$ excludes the line graphs of all subdivisions of some wall, then the treewidth of every graph in $\mathcal{C}$ is at most a polynomial function of its clique number. This is best possible in that both (a) and (b) are necessary for the existence of $any$ function with the above property.