Spanning trees and the complexity of flood-filling games

Meeks, K. and Scott, A. (2014) Spanning trees and the complexity of flood-filling games. Theory of Computing Systems, 54(4), pp. 731-753. (doi: 10.1007/s00224-013-9482-z)

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.

• Item Type: Articles
• Additional Information: The final publication is available at Springer via http://dx.doi.org/10.1007/s00224-013-9482-z
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Meeks, Dr Kitty
• Authors: Meeks, K., and Scott, A.
• Subjects: Q Science > QA Mathematics; Q Science > QA Mathematics > QA75 Electronic computers. Computer science
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Theory of Computing Systems
• Publisher: Springer Verlag
• ISSN: 1432-4350
• ISSN (Online): 1433-0490
• Copyright Holders: Copyright © 2013 Springer Science+Business Media New York
• First Published: First published in Theory of Computing Systems 54(4):731-753
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher