Counterexamples regarding linked and lean tree-decompositions of infinite graphs
arXiv:2405.06755
Abstract
Kriz and Thomas showed that every (finite or infinite) graph of tree-width admits a lean tree-decomposition of width . 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.
Revision based on referee comments