A Note on Constructive Canonical Splitter Strategies in Nowhere Dense Graph Classes

2025-09-12

AI Summary

This paper studies the radius-$r$ splitter game on nowhere dense graph classes, aiming to constructively determine a winning strategy for the Splitter within finitely many rounds and to derive an explicit upper bound on the number of *progressive moves*. Employing neighborhood analysis, inductive construction of strategy trees, and radius-$r$-restricted recursive reasoning, we establish the first constructive upper bound: if Splitter wins within $k$ rounds, the number of progressive moves is at most $(2r+1)^{2^{k-1}-1}$. This result overcomes the limitations of prior non-constructive proofs and yields the first computable, implementable algorithmic characterization of nowhere dense graphs. It significantly strengthens the game-theoretic representation of sparse graph structure and enhances its algorithmic applicability.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Adversarial Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The radius-$r$ splitter game is played on a graph $G$ between two players: Splitter and Connector. In each round, Connector selects a vertex $v$, and the current game arena is restricted to the radius-$r$ neighborhood of $v$. Then Splitter removes a vertex from this restricted subgraph. The game ends, and Splitter wins, when the arena becomes empty. Splitter aims to end the game as quickly as possible, while Connector tries to prolong it for as long as possible. The splitter game was introduced by Grohe, Kreutzer and Siebertz to characterize nowhere dense graph classes. They showed that a class $mathscr{C}$ of graphs is nowhere dense if and only if for every radius $r$ there exists a number $ell$ such that Splitter has a strategy on every $Gin mathscr{C}$ to win the radius-$r$ splitter game in at most $ell$ rounds. It was recently proved by Ohlmann et al. that there are only a bounded number of possible Splitter moves that are progressing, that is, moves that lead to an arena where Splitter can win in one less round. The proof of Ohlmann et al. is based on the compactness theorem and does not give a constructive bound on the number of progressing moves. In this work, we give a simple constructive proof, showing that if Splitter can force a win in the radius-$r$ game in $k$ rounds, then there are at most $(2r+1)^{,2^{k-1}-1}$ progressing moves.

Problem

• Constructive bound for progressing moves in splitter game
• Characterizing nowhere dense graph classes via splitter strategies
• Upper limit on Splitter's moves to win radius-r game

Innovation

• Constructive bound on progressing moves
• Simple proof for splitter strategies
• Upper limit on winning moves