polish—Let us play the cleaning game

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 Kn provided n≥3 and the complete bipartite graph K2,n provided n≠1,3. 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.