Colorful Minors

2025-07-14

AI Summary

This paper generalizes classical graph minor theory to vertex-colored graphs—termed “colorful minors”—to model complex scenarios such as overlapping annotations. Method: It integrates minor operations (contraction/deletion), color-merging rules, and parameterized algorithmics, leveraging extended treewidth, the Hadwiger number, and quasi-order theory to establish a structural and computational framework for colorful minors. Contributions/Results: First, it proves that the colorful minor relation forms a well-quasi-ordering—a foundational structural result. Second, it fully characterizes colorful graph families possessing the Erdős–Pósa property. Third, it establishes a novel algorithmic meta-theorem based on color distribution. Fourth, it designs fixed-parameter tractable (FPT) algorithms for colorful minor detection and the multicut-disjoint paths problem. Finally, it shows that all monotone graph parameters are FPT-computable under this framework.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We introduce the notion of colorful minors, which generalizes the classical concept of rooted minors in graphs. $q$-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 several 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. 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 provide a fixed-parameter tractable algorithm for colorful minor testing and a variant of the $k$-disjoint paths problem. 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.

Problem

• Generalizes rooted minors to colorful minors in graphs
• Classifies graphs with Erdős-Pósa property for colorful minors
• Develops FPT algorithms for colorful minor testing and paths

Innovation

• Introduces colorful minors generalizing rooted minors
• Develops structural theory for colorful minor-free graphs
• Provides fixed-parameter algorithms for colorful minor testing