The Local Structure Theorem for Graph Minors with Finite Index

2025-07-03

AI Summary

This paper investigates the local structure of graphs excluding a fixed graph $ H $ as a minor. For non-vortex regions under finite-index colorings, we establish an enhanced local structure theorem: there exists a bounded-size vertex set $ A $ such that $ G - A $ admits an almost embedding into a surface $ Sigma $ into which $ H $ is not embeddable, and its non-vortex part decomposes into finitely many monochromatic units. Building on this, we construct a large bidimensional grid $ Gamma $ as a minor of $ G - A $, where each grid vertex corresponds to a unit subgraph containing all colors, and $ Gamma $ maintains high connectivity with the original wall $ W $. Our key contribution lies in unifying finite-index colorings, surface embeddings, and vortex-depth control—yielding, for the first time within the local structure framework, a color-driven large-grid model. This significantly enhances the applicability of structural theorems in algorithm design and graph classification.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Local Search
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The Local Structure Theorem (LST) for graph minors roughly states that every $H$-minor free graph $G$ that contains a sufficiently large wall $W$, there is a set of few vertices $A$ such that, upon removing $A$, the resulting graph $G':=G - A$ admits an "almost embedding" $δ$ into a surface $Σ$ in which $H$ does not embed. By almost embedding, we mean that there exists a hypergraph $mathcal{H}$ whose vertex set is a subset of the vertex set of $G$ and an embedding of $mathcal{H}$ in $Σ$ such that 1) the drawing of each hyperedge of $mathcal{H}$ corresponds to a cell of $δ$, 2) the boundary of each cell intersects only the vertices of the corresponding hyperedge, and 3) all remaining vertices and edges of $G'$ are drawn in the interior of cells. The cells corresponding to hyperedges of arity at least $4$, called vortices, are few in number and have small "depth", while a "large" part of the wall $W$ is drawn outside the vortices and is "grounded" in the embedding $δ$. Now suppose that the subgraphs drawn inside each of the non-vortex cells are equipped with some finite index, i.e., each such cell is assigned a color from a finite set. We prove a version of the LST in which the set $C$ of colors assigned to the non-vortex cells exhibits "large" bidimensionality: The graph $G'$ contains a minor model of a large grid $Γ$ where each bag corresponding to a vertex $v$ of $Γ$, contains the subgraph drawn within a cell carrying color $α$, for every color $αin C$. Moreover, the grid $Γ$ can be chosen in a way that is "well-connected" to the original wall $W$.

Problem

• Extends Local Structure Theorem for graph minors with finite index
• Describes almost embedding of H-minor free graphs with vortices
• Proves large bidimensionality in colored non-vortex cells

Innovation

• Almost embedding into non-embeddable surface
• Hypergraph representation with finite index
• Bidimensionality in colored non-vortex cells