Bounds on the Twin-Width of Product Graphs

TL;DR

This paper investigates how the twin-width parameter behaves under various graph products, establishing bounds for many types and showing limitations for the modular product, with tightness examples and improved bounds for specific graph classes.

Contribution

It extends the understanding of twin-width bounds across multiple graph product types, including new bounds for several products and demonstrating the non-existence of bounds for the modular product.

Findings

Bounds established for Cartesian, tensor, corona, rooted, replacement, and zig-zag products.

Twin-width of lexicographical product equals the maximum of individual twin-widths.

No bound exists for the modular product's twin-width.

Abstract

Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomass\'{e} & Watrigant. Given two graphs $G$ and $H$ and a graph product $\star$, we address the question: is the twin-width of $G\star H$ bounded by a function of the twin-widths of $G$ and $H$ and their maximum degrees? It is known that a bound of this type holds for strong products (Bonnet, Geniet, Kim, Thomass\'{e} & Watrigant; SODA 2021). We show that bounds of the same form hold for Cartesian, tensor/direct, corona, rooted, replacement, and zig-zag products. For the lexicographical product it is known that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs (Bonnet, Kim, Reinald, Thomass\'{e} & Watrigant; IPEC 2021). In contrast, for the modular product we show that no bound can hold. In addition, we provide examples showing many of our bounds are tight, and give improved bounds for certain classes of graphs.