分裂图子类上黄金抓取问题的贪心方法
Greedy approaches for Gold Grabbing on subclasses of split graphs
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中文总结 AI 辅助
该研究针对分裂图子类上的黄金抓取游戏，分析贪心策略的最优性，证明其对一般分裂图非最优，但对完全分裂图$CS_{(2,n)}$可最大化游戏价值，且顶点数为偶数时先手不败。
中文摘要 AI 辅助
黄金抓取游戏是一种在顶点赋权图上进行的组合游戏，两名玩家轮流移除顶点，同时保持图的连通性，目标是最大化收集的总权重。尽管现有文献主要关注能保证胜利的策略，但最优性问题（即最大化总收益）仍较少被研究。本工作研究该场景下贪心策略的表现，证明该方法对一般分裂图并非最优，凸显该类图的结构局限性；另一方面，证明对于完全分裂图$CS_{(2,n)}$，贪心策略的移动序列可最大化游戏价值，因此当顶点数为偶数时先手不会输。这些结果有助于更好地理解确保基于图的组合游戏中简单策略最优性的结构条件。
英文摘要
The Gold Grabbing Game is a combinatorial game on vertex-weighted graphs in which two players alternately remove vertices while maintaining graph connectivity, aiming to maximize the total collected weight. Although the literature has primarily focused on strategies that guarantee victory, the question of optimality --- i.e., maximizing total gain --- remains less explored. In this work, we investigate the behavior of the greedy strategy in this setting. We show that this approach is not optimal for general split graphs, highlighting structural limitations of this class. On the other hand, we prove that for complete split graphs $CS_{(2,n)}$, the greedy strategy yields a sequence of moves that maximizes the game value. As a consequence, the first player does not lose when the number of vertices is even. These results contribute to a better understanding of the structural conditions that ensure the optimality of simple strategies in graph-based combinatorial games.
