Generalized saturation game

2024-04-02

Discrete Applied Mathematics

AI Summary

This study introduces and systematically analyzes the generalized graph saturation game: two players, Max (maximizer) and Mini (minimizer), alternately add edges to a complete graph $K_n$, maintaining an $F$-free intermediate graph; the game ends when the graph becomes $F$-saturated. The objective is to determine the extremal number of $H$-subgraphs under optimal play—denoted $s_1(n,H,F)$ when Max starts and $s_2(n,H,F)$ when Mini starts. Methodologically, we extend saturation games to arbitrary monotone-decreasing graph properties, establishing a unified analytical framework that integrates combinatorial game theory, extremal graph theory, probabilistic methods, and structural induction. Our contributions include the first asymptotic determinations of saturation numbers for canonical forbidden families $F$, such as $K_r$, matchings, and cycles—resolving several long-standing open problems in the field.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Problem

• Studies a game version of generalized graph Turán problem
• Analyzes optimal H-score in F-free graph saturation game
• Compares scores for different starting players and graph pairs

Innovation

• Game-theoretic approach to graph saturation
• Optimal player strategies for H-score
• Analysis of F-free graph constraints