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A graph is O k -free if it does not contain k pairwise vertex-disjoint and non-adjacent cycles. We prove that "sparse" (here, not containing large complete bipartite graphs as subgraphs) O k -free graphs have treewidth (even, feedback vertex set number) at most logarithmic in the number of vertices. This is optimal, as there is an infinite family of O 2 -free graphs without K 2 , 3 as a subgraph and whose treewidth is (at least) logarithmic. Using our result, we show that Maximum Independent Set and 3-Coloring in O k -free graphs can be solved in quasi-polynomial time. Other consequences include that most of the central NP-complete problems (such as Maximum Independent Set , Minimum Vertex Cover , Minimum Dominating Set , Minimum Coloring) can be solved in polynomial time in sparse O k -free graphs, and that deciding the O k -freeness of sparse graphs is polynomial time solvable. [ABSTRACT FROM AUTHOR]

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., 2022; Przybyszewski, 2022) 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. (2022), 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. [ABSTRACT FROM AUTHOR]