We introduce the notion of colorful minors, which generalizes the classical concept of rooted minors in graphs. A $q$-colorful graph is defined as a pair $(G, χ),$ where $G$ is a graph and $χ$ assigns to each vertex a (possibly empty) subset of at most $q$ colors. The colorful minor relation enhances the classical minor relation by merging color sets at contracted edges and allowing the removal of colors from vertices. This framework naturally models algorithmic problems involving graphs with (possibly overlapping) annotated vertex sets. We develop a structural theory for colorful minors by establishing three core theorems characterizing $\mathcal{H}$-colorful minor-free graphs, where $\mathcal{H}$ consists either of a clique or a grid with all vertices assigned all colors, or of grids with colors segregated and ordered on the outer face. Our results reveal that when exclusion is imposed not only on graphs but also to the way colors are distributed in them, a more refined structural landscape appears. Leveraging our structural insights, we provide a complete classification -- parameterized by the number $q$ of colors -- of all colorful graphs that exhibit the Erdős-Pósa property with respect to colorful minors. On the algorithmic side, we deduce that colorful minor testing is fixed-parameter tractable. Together with the fact that the colorful minor relation forms a well-quasi-order, this implies that every colorful minor-monotone parameter on colorful graphs admits a fixed-parameter algorithm. Furthermore, we derive two algorithmic meta-theorems (AMTs) whose structural conditions are linked to extensions of treewidth and Hadwiger number on colorful graphs. Our results suggest how known AMTs can be extended to incorporate not only the structure of the input graph but also the way the colored vertices are distributed in it.

翻译：我们引入了彩色子式的概念，它推广了图中经典的有根子式概念。一个 $q$-彩色图定义为一个二元组 $(G, χ)$，其中 $G$ 是一个图，而 $χ$ 为每个顶点分配一个（可能为空的）至多包含 $q$ 种颜色的子集。彩色子式关系通过合并收缩边上的颜色集并允许从顶点移除颜色，增强了经典的子式关系。该框架自然地建模了涉及具有（可能重叠的）标注顶点集的图的算法问题。我们通过建立三个核心定理来发展彩色子式的结构理论，这些定理刻画了 $\mathcal{H}$-彩色子式自由图，其中 $\mathcal{H$ 要么是一个所有顶点都被分配了所有颜色的团或网格，要么是颜色在外围面上被隔离并有序排列的网格。我们的结果表明，当排除不仅施加于图本身，还施加于图中颜色的分布方式时，会出现一个更精细的结构图景。利用我们的结构洞见，我们提供了一个完整的分类——以颜色数量 $q$ 为参数——关于所有关于彩色子式具有 Erdős-Pósa 性质的彩色图。在算法方面，我们推断彩色子式检测是固定参数可处理的。结合彩色子式关系构成一个良拟序这一事实，这意味着彩色图上的每一个彩色子式单调参数都允许一个固定参数算法。此外，我们推导出两个算法元定理，其结构条件与彩色图上树宽和 Hadwiger 数的扩展相关联。我们的结果表明了如何将已知的算法元定理进行扩展，以不仅纳入输入图的结构，还纳入着色顶点在其中分布的方式。