Linear-time implementation of the last-vertex recognition scheme

Establish whether the recognition scheme that guesses the last vertex to become incident to a red edge can be implemented in linear time with further arguments.

Background

The paper develops a linear-time recognition algorithm for graphs of twin-width at most 1 using structural decomposition and permutation diagrams. It also discusses an alternative scheme that avoids enumerating possible first contractions by guessing the vertex contracted last; the authors reduce the candidates to four but do not settle whether that specific scheme itself can be implemented in linear time.

References

However, we do not exclude the possibility that the scheme of the previous algorithm can be implemented in linear time with some further arguments.

— Twin-width one  (2501.00991 - Ahn et al., 2 Jan 2025) in Section 1, Introduction, paragraph beginning “Regarding the challenge of avoiding…”