Neighbourhood complexity of graphs of bounded twin-width - Archive ouverte HAL Access content directly
Journal Articles European Journal of Combinatorics Year : 2024

Neighbourhood complexity of graphs of bounded twin-width


We give essentially tight bounds for, ν(d, k), the maximum number of distinct neighbourhoods on a set X of k vertices in a graph with twin-width at most d. Using the celebrated Marcus-Tardos theorem, two independent works [Bonnet et al., Algorithmica '22; Przybyszewski '22] have shown the upper bound ν(d, k) ⩽ exp(exp(O(d)))k, with a double-exponential dependence in the twin-width. The work of [Gajarsky et al., ICALP '22], using the framework of local types, implies the existence of a single-exponential bound (without explicitly stating such a bound). We give such an explicit bound, and prove that it is essentially tight. Indeed, we give a short self-contained proof that for every d and k ν(d, k) ⩽ (d + 2)2 d+1 k = 2 d+log d+Θ(1) k, and build a bipartite graph implying ν(d, k) ⩾ 2 d+log d+Θ(1) k, in the regime when k is large enough compared to d.
Fichier principal
Vignette du fichier
Twin_width_neighbourhood_complexity.pdf (416.64 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-04177614 , version 1 (05-08-2023)



Édouard Bonnet, Florent Foucaud, Tuomo Lehtilä, Aline Parreau. Neighbourhood complexity of graphs of bounded twin-width. European Journal of Combinatorics, 2024, 115, pp.103772. ⟨10.1016/j.ejc.2023.103772⟩. ⟨hal-04177614⟩
303 View
33 Download



Gmail Mastodon Facebook X LinkedIn More