A generalization of Arc-Kayles

TL;DR

This paper introduces Weighted Arc Kayles, a generalization of Arc-Kayles played on vertex-counter graphs, providing strategies for trees of depth 2, analyzing Grundy values, and exploring periodicity and connections to chess variants.

Contribution

It develops a winning strategy for WAK on certain trees, analyzes its Grundy values, and establishes periodicity and links to chess variants, advancing understanding of this generalized game.

Findings

Winning strategy for WAK on trees of depth 2

Grundy values of WAK and Arc-Kayles are unbounded

Periodic outcome behavior when counters are fixed except at one vertex

Abstract

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.