First-order model checking for flip-width

Determine the parameterized complexity of first-order model-checking on graph classes of bounded flip-width.

Background

Flip-width is described as a more general parameter than twin-width and is known to admit an XP approximation algorithm. The paper notes that, despite this approximation result, the status of first-order model-checking for bounded flip-width is unresolved.

References

Interestingly, the more general parameter flip-width introduced by Torunczyk has an XP approximation algorithm but the status of FO model-checking for this parameter is unknown.

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