Counterexamples regarding linked and lean tree-decompositions of infinite graphs

TL;DR

This paper explores the limitations of extending Kriz and Thomas's result on lean tree-decompositions from finite to infinite graphs, providing counterexamples that highlight these boundaries.

Contribution

It constructs specific counterexamples, including a locally finite, planar, connected graph without lean tree-decomposition, showing the limits of generalizing existing theorems.

Findings

Counterexamples demonstrate limits of extending lean tree-decomposition to infinite graphs.

A locally finite, planar, connected graph with no lean tree-decomposition is constructed.

The results highlight fundamental differences between finite and infinite graph decompositions.

Abstract

Kriz and Thomas showed that every (finite or infinite) graph of tree-width $k \in \mathbb{N}$ admits a lean tree-decomposition of width $k$. We discuss a number of counterexamples demonstrating the limits of possible generalisations of their result to arbitrary infinite tree-width. In particular, we construct a locally finite, planar, connected graph that has no lean tree-decomposition.