Twin-width and types
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We study problems connected to first-order logic in graphs of bounded twin-width. Inspired by the approach of Bonnet et al. [FOCS 2020], we introduce a robust methodology of local types and describe their behavior in contraction sequences – the decomposition notion underlying twin-width. We showcase the applicability of the methodology by proving the following two algorithmic results. In both statements, we fix a first-order formula φ(x_1,…,x_k) and a constant d, and we assume that on input we are given a graph G together with a contraction sequence of width at most d. (A) One can in time O(n) construct a data structure that can answer the following queries in time O(loglog n): given w_1,…,w_k, decide whether ϕ(w_1,…,w_k) holds in G. (B) After O(n)-time preprocessing, one can enumerate all tuples w_1,…,w_k that satisfy ϕ(x_1,…,x_k) in G with O(1) delay. In the case of (A), the query time can be reduced to O(1/ε) at the expense of increasing the construction time to O(n^1+ε), for any fixed ε>0. Finally, we also apply our tools to prove the following statement, which shows optimal bounds on the VC density of set systems that are first-order definable in graphs of bounded twin-width. (C) Let G be a graph of twin-width d, A be a subset of vertices of G, and φ(x_1,…,x_k,y_1,…,y_l) be a first-order formula. Then the number of different subsets of A^k definable by ϕ using l-tuples of vertices from G as parameters, is bounded by O(|A|^l).
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