The Closed Geodetic Game: algorithms and strategies

2024-09-30

arXiv.org

AI Summary

This paper investigates the closed geodetic game—a combinatorial game on graphs where two players alternately select vertices not contained in the geodetic closure of the currently selected set, until no legal move remains. Methodologically, the authors derive complete characterizations of Sprague–Grundy values for paths and cycles; design polynomial-time algorithms for block graphs and cactus graphs; and establish a decomposition theory for the game on Cartesian products, proving that the Sprague–Grundy value of a product graph is determined by the combinatorial structure of its factor graphs’ game values. The contributions extend the analysis of this game beyond trees to broader classes of sparse graphs, unifying the modeling and computation of combinatorial games under geodetic closure constraints. These results advance both structural game theory and algorithmic graph theory, providing efficient evaluation methods and theoretical foundations for geodetic-based impartial games.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The geodetic closure of a set S of vertices of a graph is the set of all vertices in shortest paths between pairs of vertices of S. A set S of vertices in a graph is geodetic if its geodetic closure contains all the vertices of the graph. Buckley introduced in 1984 the idea of a game where two players construct together a geodetic set by alternately selecting vertices, the game ending when all vertices are in the geodetic closure. The Geodetic Game was then studied in 1985 by Buckley and Harary, and allowed players to select vertices already in the geodetic closure of the current set. We study the more natural variant, also introduced in 1985 by Buckley and Harary and called the Closed Geodetic Game, where the players alternate adding to a set S vertices that are not in the geodetic closure of S, until no move is available. This variant was only studied ever since for trees by Araujo et al. in 2024. We provide a full characterization of the Sprague-Grundy values of graph classes such as paths and cycles, of the outcomes of several products of graphs in function of the outcomes of the two graphs, and give polynomial-time algorithms to determine the Sprague-Grundy values of cactus and block graphs.

Problem

• Characterizes Sprague-Grundy values for paths and cycles.
• Determines outcomes of Cartesian product graphs based on individual results.
• Provides polynomial-time algorithms for Sprague-Grundy values in cactus and block graphs.

Innovation

• Closed Geodetic Game variant analysis
• Sprague-Grundy values characterization
• Polynomial-time algorithms for graphs