Recognizing graphs of twin-width two and three

Determine whether recognizing graphs of twin-width at most 2 and graphs of twin-width at most 3 admits a polynomial-time algorithm, thereby resolving the computational complexity of these recognition problems.

Background

The paper contrasts the known polynomial-time recognition algorithm for graphs of twin-width at most 1 with the substantially less understood cases of twin-width 2 and 3. Although distinguishing twin-width 4 from twin-width 5 is NP-hard, the corresponding complexity of recognizing graphs with twin-width at most 2 or 3 is not settled in the cited discussion.

References

Meanwhile, the hardness of recognizing twin-width~$2$ and~$3$ graphs remains open.

— Twin-width one  (2501.00991 - Ahn et al., 2 Jan 2025) in Section 1, Introduction, paragraph beginning “Twin-width is an invariant…”