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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)
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