FO model-checking on bounded flip-width classes

Determine whether First-Order model-checking is fixed-parameter tractable on graph classes of bounded flip-width, even when suitable Flipper strategies are supplied through oracles.

Background

Flip-width is proposed as a dense analogue of bounded expansion and is defined using a game in which Flipper repeatedly applies boundedly many flips. The survey establishes that bounded flip-width is an FO ideal and contains bounded-expansion and bounded-twin-width classes, but no model-checking theorem is known for it.

References

It is unknown whether the $FO$ model-checking problem can be solved in fixed-parameter time on classes of bounded flip-width, even under the assumption of being provided suitable Flipper's strategies through oracles.

— Graph classes through the lens of logic  (2501.04166 - Pilipczuk, 7 Jan 2025) in Section 6, paragraph “Flip-width and dense analogues of sparse notions”

In particular, it is not known whether the model checking results for classes of bounded expansion and for classes of bounded twin-width (with a given contraction sequence) extend to classes of bounded flip-width.

— Merge-width and First-Order Model Checking  (2502.18065 - Dreier et al., 25 Feb 2025) in Section 1, paragraph “Flip-width”