Bounds on the Twin-Width of Product Graphs

Abstract

Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomassé & Watrigant. Given two graphs G and H and a graph product ⋆, we address the question: is the twin-width of G ⋆ 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, Thomassé & Watrigant; SODA 2021).We show that bounds of the same form hold for Cartesian, tensor/direct, rooted, replacement, and zig-zag products. For the lexicographical product we prove that the twin-width of the product of two graphs is exactly the maximum of the twin-widths of the individual graphs. 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.

Twin-width can be exponential in treewidth

“…Despite their apparent generality, classes of bounded twin-width are small [7], χ-bounded [8], even quasi-polynomially χ-bounded [17], preserved (albeit with a higher upper bound) by first-order transductions [12], and by the usual graph products when one graph has bounded degree [16,7], have VC density 1 [11,19], admit, when O(1)-sequences are given, a fixed-parameter tractable first-order model checking [12], an (almost) single-exponential parameterized algorithm for various problems that are W[1]-hard in general [8], as well as a parameterized fully-polynomial linear algorithm for counting triangles [15], an (almost) linear representation [18], a stronger regularity lemma [19], etc.…”