The complexity of Free-Flood-It on 2×n boards

We consider the complexity of problems related to the combinatorial game Free-Flood-It, in which players aim to make a coloured graph monochromatic with the minimum possible number of flooding operations. Our main result is that computing the length of an optimal sequence is fixed parameter tractable (with the number of colours as a parameter) when restricted to rectangular 2×n boards. We also show that, when the number of colours is unbounded, the problem remains NP-hard on such boards. These results resolve a question of Clifford, Jalsenius, Montanaro and Sach.