Approximating twin-width is FPT parameterized by treedepth via oriented twin-width, and exact twin-width computation is FPT parameterized by vertex integrity.
16 Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant
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Computing Twin-Width via Treedepth and Vertex Integrity
Approximating twin-width is FPT parameterized by treedepth via oriented twin-width, and exact twin-width computation is FPT parameterized by vertex integrity.