Compact representation for matrices of bounded twin-width

Pilipczuk, Michał; Sokołowski, Marek; Zych-Pawlewicz, Anna

doi:10.48550/arxiv.2110.08106

Cited by 1 publication

(1 citation statement)

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“…In particular, twin-width is bounded on some classes of dense graphs (e.g., the class of complete graphs has twin-width zero) and on some classes of sparse graphs (e.g., the class of path graphs has twin-width one), and indeed both graph classes of bounded tree-width and graph classes of bounded rank-width have bounded twin-width [12,24]. Several other aspects of twin-width, such as χ-boundedness, colouring numbers and VC-density, have recently received attention [10,15,31,32,33].…”

Section: Introductionmentioning

confidence: 99%

Bounds on the Twin-Width of Product Graphs

Pettersson¹,

Sylvester²

2022

Preprint

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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.

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