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

TL;DR

This paper demonstrates that every connected graph has a tree decomposition indexed by a subgraph of the graph itself, with width bounded by a function of the graph's pathwidth, using Simon's Factorization Theorem.

Contribution

It introduces a novel application of Simon's Factorization Theorem to construct specific tree decompositions with subgraph index trees, answering a previously open question.

Findings

Existence of such tree decompositions for all connected graphs

Width bounded by a function of the graph's pathwidth

Simplified proof illustrating the technique's utility

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.