The complexity of flood-filling games on graphs

Meeks, K. and Scott, A. (2012) The complexity of flood-filling games on graphs. Discrete Applied Mathematics, 160(7-8), pp. 959-969. (doi: 10.1016/j.dam.2011.09.001)

Abstract

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. Although computing the minimum number of moves required to flood an arbitrary graph is known to be NP-hard, we demonstrate a polynomial time algorithm to compute the minimum number of moves required to link each pair of vertices. We apply this result to compute in polynomial time the minimum number of moves required to flood a path, and an additive approximation to this quantity for an arbitrary k × n board, coloured with a bounded number of colours, for any fixed k . On the other hand, we show that, for k ≥ 3 , determining the minimum number of moves required to flood a k × n board coloured with at least four colours remains NP-hard.

• Item Type: Articles
• Additional Information: NOTICE: this is the author’s version of a work that was accepted for publication in Discrete Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Discrete Applied Mathematics 160(7-8):959-969 October 2011 DOI: 10.1016/j.dam.2011.09.001
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Meeks, Dr Kitty
• Authors: Meeks, K., and Scott, A.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Discrete Applied Mathematics
• Publisher: Elsevier B.V.
• ISSN: 0166-218X
• ISSN (Online): 1872-6771
• Copyright Holders: Copyright © 2011 Elsevier B.V.
• First Published: First published in Discrete Applied Mathematics 160(7-8):959-969
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher