Flip-width and structurally bounded expansion

Determine whether edge-stable graph classes of bounded flip-width are exactly the structurally bounded expansion classes.

Background

Structurally bounded expansion classes are obtained by FO-transducing bounded-expansion classes. The survey notes that bounded flip-width is another candidate dense analogue of bounded expansion and asks whether imposing edge-stability makes the two notions coincide.

References

It is unknown whether edge-stable classes of bounded flip-width coincide with classes of structurally bounded expansion.

— 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”