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 $k$ such that Splitter has a strategy on every $G\in \mathscr{C}$ to win the radius-$r$ splitter game in at most $k$ rounds. It was recently proved by Ohlmann et al. that for every nowhere dense class $\mathscr{C}$ and every radius $r$ 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.

翻译：半径-$r$分割博弈在两个玩家之间进行：分割者与连接者。每轮中，连接者选择一个顶点$v$，当前博弈区域被限制在$v$的半径-$r$邻域内。随后分割者从该受限子图中移除一个顶点。当区域变为空时博弈结束，分割者获胜。分割者旨在尽快结束博弈，而连接者则试图尽可能延长博弈。分割博弈由Grohe、Kreutzer和Siebertz引入，用于刻画无处稠密图类。他们证明：图类$\mathscr{C}$是无处稠密的，当且仅当对每个半径$r$存在一个数$k$，使得分割者在每个$G\in \mathscr{C}$上都有策略在至多$k$轮内赢得半径-$r$分割博弈。Ohlmann等人最近证明，对每个无处稠密类$\mathscr{C}$和每个半径$r$，可能的分割者前进走法（即导致分割者能少一轮获胜的区域）数量有界。Ohlmann等人的证明基于紧致性定理，未给出前进走法数量的构造性界。本文给出一个简单的构造性证明，表明若分割者能在$k$轮内强制赢得半径-$r$博弈，则前进走法的数量至多为$(2r+1)^{\,2^{k-1}-1}$。