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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]

For any small positive real ε and integer t > 1 ε , we build a graph with a vertex deletion set of size t to a tree, and twin-width greater than 2 (1 − ε) t. In particular, this shows that the twin-width is sometimes exponential in the treewidth, in the so-called oriented twin-width and grid number, and that adding an apex may multiply the twin-width by at least 2 − ε. Except for the one in oriented twin-width, these lower bounds are essentially tight. [ABSTRACT FROM AUTHOR]