# 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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