Playing Games with Cacti

TL;DR

This paper analyzes the Game of Cycles played on cactus graphs, extending previous symmetry-based strategies with a modified mirror-reverse approach to determine winning strategies.

Contribution

It introduces a modified mirror-reverse strategy for cactus graphs, advancing understanding of winning strategies in the Game of Cycles.

Findings

Identifies winning strategies for specific cactus graphs.

Extends symmetry-based methods to a new class of graphs.

Provides insights into game dynamics on planar graphs.

Abstract

The Game of Cycles is a two-player impartial mathematical game, introduced by Francis Su in his book Mathematics for Human Flourishing (2020). The game is played on simple planar graphs in which players take turns marking edges using a sink-source rule. In Alvarado et al., the authors determine who is able to win on graphs with certain types of symmetry using a mirror-reverse strategy. In this paper, we analyze the game for specific types of cactus graphs using a modified version of the mirror-reverse strategy.