The Game of Arrows on 3-Legged Spider Graphs

TL;DR

This paper analyzes the Game of Arrows, a variant of the Game of Cycles, on 3-legged spider graphs, revealing that player two always wins on odd-length legs using the Sprague-Grundy Theorem.

Contribution

It determines the winning strategy for the Game of Arrows on infinite families of asymmetric spider graphs, extending previous symmetry-based results.

Findings

Player two has a winning strategy on any 3-legged spider with odd-length legs.

The Sprague-Grundy Theorem is used to analyze game states and compute Grundy values.

The game is simplified by removing the cycle cell condition for trees, called the Game of Arrows.

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.