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Deciding twin-width at most 4 is NP-complete


We show that determining if an n-vertex graph has twin-width at most 4 is NP-complete, and requires time 2 Ω(n/ log n) unless the Exponential-Time Hypothesis fails. Along the way, we give an elementary proof that n-vertex graphs subdivided at least 2 log n times have twin-width at most 4. We also show how to encode trigraphs H (2-edge colored graphs involved in the definition of twin-width) into graphs G, in the sense that every d-sequence (sequence of vertex contractions witnessing that the twin-width is at most d) of G inevitably creates H as an induced subtrigraph, whereas there exists a partial d-sequence that actually goes from G to H. We believe that these facts and their proofs can be of independent interest.
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hal-03750997 , version 1 (13-08-2022)



Pierre Bergé, Édouard Bonnet, Hugues Déprés. Deciding twin-width at most 4 is NP-complete. 49th EATCS International Colloquium on Automata, Languages and Programming (ICALP 2022), Jul 2022, Paris, France. ⟨10.4230/LIPIcs.ICALP.2022.18⟩. ⟨hal-03750997⟩
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