Deciding twin-width at most 4 is NP-complete

Pierre Bergé; Édouard Bonnet; Hugues Déprés

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arXiv · 2112.08953

Deciding twin-width at most 4 is NP-complete

Pierre Bergé·Édouard Bonnet·Hugues Déprés

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Abstract

We show that determining if an $n$-vertex graph has twin-width at most 4 is NP-complete, and requires time $2^{\Omega(n/\log n)}$ 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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BibTeX RIS

Pierre Bergé, Édouard Bonnet, Hugues Déprés. 2021-12-16. Deciding twin-width at most 4 is NP-complete. https://arxiv.org/abs/2112.08953

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