The Game of Arrows on 3-Legged Spider Graphs

Abstract: The Game of Cycles is a combinatorial game introduced by Francis Su in 2020 in which players take turns marking arrows on the edges of a simple plane graph, avoiding the creation of sinks and sources and seeking to complete a "cycle cell." Su and his collaborators (2021) found winning strategies on graphs with certain types of symmetry using reverse mirroring. In this paper, we for the first time determine the winning player in the Game of Cycles on an infinite family of graphs lacking symmetry. In particular, we use the Sprague-Grundy Theorem to show that player two has a winning strategy for the Game of Cycles on any 3-legged spider graph with legs of odd length. Because the cycle cell victory condition is extraneous for tree graphs (including spiders), we drop it from the rules and call the result the Game of Arrows. Our proof leans heavily on a notion of state isomorphism that allows us to decompose a game state into states of smaller pieces of a graph, leading to nim-sum calculations with Grundy values.