Twin-width can be exponential in treewidth
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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.
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