Blow-up structure of graphs excluding a tree or an apex-tree as a minor

2026-03-17

AI Summary

This study investigates the structural properties of graph classes excluding either an arbitrary $t$-vertex tree $T$ or its apex-tree extension $T^+$ as a minor. By employing graph product decomposition techniques, the authors embed such graphs into Cartesian products of low-pathwidth (or low-treewidth) host graphs and complete graphs. Specifically, graphs excluding $T$ are embedded into the product of a graph of pathwidth at most $2h - 1$ and $K_{t-2}$, while those excluding $T^+$ are embedded into the product of a graph of treewidth at most $4h - 1$ and $K_{2(t-1)d}$. These results significantly improve upon prior bounds by reducing the order of the complete graph factor and nearly optimally lowering the treewidth bound of the host graph in the apex-tree case.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

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

Abstract

We prove blow-up structure theorems for graphs excluding a tree or an apex-tree as a minor. First, we show that for every $t$-vertex tree $T$ with $t\geq 3$ and radius $h$, and every graph $G$ excluding $T$ as a minor, there exists a graph $H$ with pathwidth at most $2h-1$ such that $G$ is contained in $H\boxtimes K_{t-2}$ as a subgraph. This improves on a recent theorem of Dujmović, Hickingbotham, Joret, Micek, Morin, and Wood (2024), who proved the same result but with a larger bound on the order of the complete graph in the product. Second, we show that for every $t$-vertex tree $T$ with $t\geq 2$, radius $h$ and maximum degree $d$, and every graph $G$ excluding the apex-tree $T^+$ as a minor, where $T^+$ is the tree obtained by adding a universal vertex to $T$, there exists a graph $H$ with treewidth at most $4h-1$ such that $G$ is contained in $H\boxtimes K_{2(t-1)d}$. The bound on the treewidth of $H$ is best possible up to a factor $2$, and improves on a $2^{h+2}-4$ bound that follows from a recent result of Dujmović, Hickingbotham, Hodor, Joret, La, Micek, Morin, Rambaud, and Wood (2024).

Problem

• graph minor
• tree
• apex-tree
• blow-up structure
• pathwidth

Innovation

• graph minor
• blow-up structure
• pathwidth
• treewidth
• apex-tree