洪水填充游戏的极值性质
Extremal properties of flood-filling games
AI总结：
本文系统研究 Flood-It 游戏中使图单色所需的最大洪水操作次数，给出任意图的上下界并证明对树紧，同时探讨高边密度图的上界改进。
摘要：
在单人组合游戏 Flood-It 中，确定使给定彩色图变为单色所需的“洪水操作”数量的问题，从算法角度已被广泛研究，但关于最坏情况下可能需要的最大移动次数的基本问题仍未得到解答。我们开始对此类问题进行系统研究，目标是确定给定图在所有可能着色下可能需要的最大移动次数。我们给出了该数量对于任意图的几个上下界，并证明所有这些界对于树都是紧的；我们还研究了如果将注意力限制在具有更高边密度的图上，上界可以改进多少。
英文摘要：
The problem of determining the number of "flooding operations" required to make a given coloured graph monochromatic in the one-player combinatorial game Flood-It has been studied extensively from an algorithmic point of view, but basic questions about the maximum number of moves that might be required in the worst case remain unanswered. We begin a systematic investigation of such questions, with the goal of determining, for a given graph, the maximum number of moves that may be required, taken over all possible colourings. We give several upper and lower bounds on this quantity for arbitrary graphs and show that all of the bounds are tight for trees; we also investigate how much the upper bounds can be improved if we restrict our attention to graphs with higher edge-density.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
