Counterexamples Regarding Linked and Lean Tree-Decompositions of Infinite Graphs

Abstract

Kříž and Thomas showed that every (finite or infinite) graph of tree-width $k\in N$ 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.