Twin-width can be exponential in treewidth

TL;DR

This paper demonstrates that twin-width can grow exponentially relative to treewidth, providing new lower bounds and insights into the relationship between these graph parameters.

Contribution

The authors construct graphs showing twin-width can be exponential in treewidth, establishing tight lower bounds and exploring effects of apex addition.

Findings

Twin-width can be exponential in treewidth.

Lower bounds are tight except for oriented twin-width.

Adding an apex can significantly increase twin-width.

Abstract

For any small positive real $\varepsilon$ and integer $t > \frac{1}{\varepsilon}$, we build a graph with a vertex deletion set of size $t$ to a tree, and twin-width greater than $2^{(1-\varepsilon) t}$. In particular, this shows that the twin-width is sometimes exponential in the treewidth, in the so-called oriented twin-width and grid number, and that adding an apex may multiply the twin-width by at least $2-\varepsilon$. Except for the one in oriented twin-width, these lower bounds are essentially tight.