Tree decompositions whose trees are subgraphs: An application of Simon's factorization

2026-02-27

AI Summary

This study addresses an open problem posed by Blanco et al.: whether one can construct a tree decomposition whose index tree is a subgraph of the original graph and whose width is governed by pathwidth rather than treewidth. To this end, we present the first construction of such a tree decomposition for any connected graph, leveraging Simon’s Factorization Theorem. We rigorously prove that the width of the resulting decomposition is bounded above by a function of the graph’s pathwidth. This work not only establishes a theoretical connection between pathwidth and this class of subgraph-indexed tree decompositions but also transcends the conventional reliance on treewidth in tree decomposition theory, offering a novel perspective for structural graph analysis.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We show that every connected graph $G$ has a tree decomposition indexed by a tree $T$ such that $T$ is a subgraph of $G$ and the width of the tree decomposition is bounded from above by a function of the pathwidth of $G$. This answers a question of Blanco, Cook, Hatzel, Hilaire, Illingworth, and McCarty (2024), who proved that it is not possible to have such a tree decomposition whose width is bounded by a function of the treewidth of $G$. The proof relies on Simon's Factorization Theorem for finite semigroups, a tool that has already been applied successfully in various areas of graph theory and combinatorics in recent years. Our application is particularly simple and can serve as a good introduction to this technique.

Problem

• tree decomposition
• subgraph
• pathwidth
• treewidth
• Simon's Factorization

Innovation

• tree decomposition
• subgraph-indexed
• pathwidth
• Simon's Factorization Theorem
• finite semigroups