幻想 figurines 翻转 Free Flood-It -- 在无 induced co-gems 图上进行多项式时间微型绘画
Fanciful Figurines flip Free Flood-It -- Polynomial-Time Miniature Painting on Co-gem-free Graphs
AI总结：
本文提出在无 induced co-gems 图上通过多项式时间算法实现微型绘画，等价于 Free Flood-It 游戏的逆过程。
摘要：
受同名爱好启发，我们引入微型绘画作为计算问题，即根据给定的模板 t:V→C 来给定图 G=(V,E) 涂色。在这一设定中，目标是使用最短的刷子笔触序列来实现模板，其中每次笔触会覆盖一个连通的顶点子集并用 C 中的一种颜色覆盖。我们证明该问题等价于研究广泛的 Free Flood-It 游戏的逆过程，其中通过最少次数将着色图变为单色。这种等价性允许将 Free Flood-It 的已知复杂性结果直接转移到微型绘画问题中，包括在严重结构限制下 NP 难的结论，例如当 G 是网格、树或分割图时。我们的主要贡献是为无 induced co-gems 图提供了一个多项式时间算法，这是一个严格扩展 cographs 的图类。作为直接结果，Free Flood-It 也在 co-gem-free 图上多项式时间可解，无论初始着色如何。
英文摘要：
Inspired by the eponymous hobby, we introduce Miniature Painting as the computational problem to paint a given graph $G=(V,E)$ according to a prescribed template $t \colon V \rightarrow C$, which assigns colors $C$ to the vertices of $G$. In this setting, the goal is to realize the template using a shortest possible sequence of brush strokes, where each stroke overwrites a connected vertex subset with a color in $C$. We show that this problem is equivalent to a reversal of the well-studied Free Flood-It game, in which a colored graph is decolored into a single color using as few moves as possible. This equivalence allows known complexity results for Free Flood-It to be transferred directly to Miniature Painting, including NP-hardness under severe structural restrictions, such as when $G$ is a grid, a tree, or a split graph. Our main contribution is a polynomial-time algorithm for Miniature Painting on graphs that are free of induced co-gems, a graph class that strictly generalizes cographs. As a direct consequence, Free Flood-It is also polynomial-time solvable on co-gem-free graphs, independent of the initial coloring.
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。
