Extremal properties of flood-filling games

The problem of determining the number of "flooding operations" required to make a given coloured graph monochromatic in the one-player combinatorial game Flood-It has been studied extensively from an algorithmic point of view, but basic questions about the maximum number of moves that might be required in the worst case remain unanswered. We begin a systematic investigation of such questions, with the goal of determining, for a given graph, the maximum number of moves that may be required, taken over all possible colourings. We give several upper and lower bounds on this quantity for arbitrary graphs and show that all of the bounds are tight for trees; we also investigate how much the upper bounds can be improved if we restrict our attention to graphs with higher edge-density.Comment: Final version, accepted to DMTC

The complexity of Free-Flood-It on 2xn 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 present as a parameter) when restricted to rectangular 2xn boards. We also show that, when the number of colours is unbounded, the problem remains NP-hard on such boards. This resolves a question of Clifford, Jalsenius, Montanaro and Sach (2010)