POLISH-Let us play the cleaning game

Authors:
Przemysaw Gordinowicz;Richard J. Nowakowski;Pawe Praat
Affiliations:
Institute of Mathematics, Technical University of Lodz, ód, Poland;Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada, B3H 3J5;Department of Mathematics, Ryerson University, Toronto, ON, Canada, M5B 2K3
Venue:
Theoretical Computer Science
Year:
2012

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Abstract

polish is a game based on the 'Cleaning with Brushes' model. It is a combinatorial game in the sense of Conway but can be seen as a graph searching or chip-firing problem as well. We consider only the impartial version and give a characterization of graphs with maximum degree two that are first player wins. We also show that the second player can win on the complete graph K"n provided n=3 and the complete bipartite graph K"2","n provided n1,3. We give the nim-values for all positions on paths, stars, and also the nim-values for complete bipartite graphs where every vertex needs at most 3 more brushes to fire.