# Metric dimension of combinatorial game graphs

## TL;DR

This paper explores the metric dimension of game graphs derived from combinatorial games, analyzing how graph-theoretic metrics can characterize game structures, including both finite and cyclic (loopy) games.

## Contribution

It introduces the concept of metric dimension in the context of combinatorial game graphs, extending graph metric analysis to both acyclic and cyclic game scenarios.

## Key findings

- Determined metric dimensions for various game graphs
- Analyzed differences between short and loopy game graphs
- Provided insights into graph-based game analysis methods

## Abstract

The study of combinatorial games is intimately tied to the study of graphs, as any game can be realized as a directed graph in which players take turns traversing the edges until reaching a sink. However, there have heretofore been few efforts towards analyzing game graphs using graph theoretic metrics and techniques. A set $S$ of vertices in a graph $G$ resolves $G$ if every vertex in $G$ is uniquely determined by the vector of its distances from the vertices in $S$. A metric basis of $G$ is a smallest resolving set and the metric dimension is the cardinality of a metric basis. In this article we examine the metric dimension of the graphs resulting from some rulesets, including both short games (those which are sure to end after finitely many turns) and loopy games (those games for which the associated graph contains cycles).