A generalization of Arc-Kayles
This work extends combinatorial game theory to weighted graphs, offering new insights for impartial games, but the results are incremental and limited to specific graph classes.
The paper introduces Weighted Arc Kayles (WAK), a generalization of Arc-Kayles where vertices have counters, and provides a winning strategy for WAK on trees of depth 2. It also shows that Grundy values for WAK and Arc-Kayles are unbounded and proves a periodicity result for WAK with fixed counters on all but one vertex.
The game Arc-Kayles is played on an undirected graph with two players taking turns deleting an edge and its endpoints from the graph. We study a generalization of this game, Weighted Arc Kayles (WAK for short), played on graphs with counters on the vertices. The two players alternate choosing an edge and removing one counter on both endpoints. An edge can no longer be selected if any of its endpoints has no counter left. The last player to play a move wins. We give a winning strategy for WAK on trees of depth 2. Moreover, we show that the Grundy values of WAK and Arc-Kayles are unbounded. We also prove a periodicity result on the outcome of WAK when the number of counters is fixed for all the vertices but one. Finally, we show links between this game and a variation of the non-attacking queens game on a chessboard.