Colorful Minors

TL;DR

The paper develops a comprehensive theory of colorful minors, extending rooted-minor concepts to q-colorful graphs and studying the interplay between color distribution and graph structure. It introduces three main structural theorems excluding rainbow clique/minors, rainbow grids, and segregated grids, with a focus on torso treewidth and near-embeddings to reveal refined decompositions when colors are distributed. It provides a full Erdős–Pósa classification for colorful graphs with respect to colorful minors, showing that reachability of the property hinges on a polynomial-sized obstruction family and that all obstructions are color-distribution aware. Algorithmically, the work proves that colorful minor testing is fixed-parameter tractable and extends Courcelle-type algorithmic meta-theorems to colorful graphs through folio-representability and CMSO/tw+dp formalisms, yielding broad applicability to annotated-graph problems. Collectively, the results illuminate how the distribution of vertex colors fundamentally shapes both structural decompositions and algorithmic tractability, enabling targeted meta-theorems and practical frameworks for colorful graph problems.

Abstract

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.

Figures (12)

• Figure 1: Two colorful graphs $(G,\chi)$ and $(H,\psi)$ such that $(H,\psi)$ is a colorful minor of $(G,\chi).$ The gray subgraphs of $(G,\chi)$ indicate the connected vertex sets that have to be contracted in order to form $(G,\psi).$ Notice that it is necessary to remove some colors from some of the vertices of $(G,\chi).$
• Figure 2: Three non-isomorphic $(4,3)$-segregated grids.
• Figure 3: The obstructions to the Erdős-Pósa property: For $q\in\mathbb{N},$ the family $\mathcal{O}_q$ consists, for $p\in[0,4],$ of the union of all $p$-colorful graphs displayed in the corresponding level above, where the levels for $p\in[4]$ are copied for each of the $\binom{q}{p}$ possible choices for $p$ out of $q$ colors.
• Figure 4: A schematic representation of the construction of a $(5,4)$-segregated grid in the proof of \ref{['thm_localSegregation']}. The striped box in the middle represents the wall $W_3$ whose bricks all represent colors $2$ and $3,$ while colors $1,$$4,$ and $5$ are connected from the outside of $W_2$ to the perimeter of $W_3$ via disjoint paths.
• Figure 5: The $15$$2$-colorful graphs from the family $\mathcal{O}_{2}^1.$ The unique $2$-colorful graph in $\mathcal{O}_{2}^1\setminus \tilde{\mathcal{O}}_{2}^1$ is highlighted in the dashed square.

Theorems & Definitions (87)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Proposition 2.1: Menger's Theorem