Polynomial-round winning strategies for bounded flip-width

Prove or refute that, in every class of bounded flip-width, the pursuer has a winning strategy terminating after a number of rounds bounded by a polynomial in the number of vertices with a fixed polynomial degree.

Background

The paper views a winning strategy for the pursuer in the flip-width game as the closest analogue of a graph decomposition. Although bounded merge-width yields strategies with linearly many rounds, it is unknown whether bounded flip-width alone guarantees strategies with polynomially bounded duration.

References

It is even unknown if in classes of bounded flip-width, the pursuer has a winning strategy -- the closest equivalent of a decomposition provided by flip-width -- which terminates after a number of rounds which is bounded by a polynomial in the number of vertices, with a fixed degree of the polynomial.

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