Spanning Trees and the Complexity of Flood-Filling Games |
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Authors: | Kitty Meeks Alexander Scott |
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Affiliation: | 1. Mathematical Institute, University of Oxford, 24-29 St. Giles, Oxford, OX1 3LB, UK
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Abstract: | We consider 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. We show that the minimum number of moves required to flood any given graph G is equal to the minimum, taken over all spanning trees T of G, of the number of moves required to flood T. This result is then applied to give two polynomial-time algorithms for flood-filling problems. Firstly, we can compute in polynomial time the minimum number of moves required to flood a graph with only a polynomial number of connected subgraphs. Secondly, given any coloured connected graph and a subset of the vertices of bounded size, the number of moves required to connect this subset can be computed in polynomial time. |
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