Deciding twin-width at most 4 is NP-complete

TL;DR

This paper proves that deciding whether a graph has twin-width at most 4 is NP-complete, providing complexity bounds and new insights into graph subdivisions and encodings related to twin-width.

Contribution

It establishes NP-completeness of twin-width at most 4 decision problem and introduces elementary proofs and encoding techniques relevant to twin-width analysis.

Findings

Deciding twin-width at most 4 is NP-complete.

Graphs subdivided at least 2 log n times have twin-width at most 4.

Encoding trigraphs into graphs reveals structural properties of twin-width.

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.