Kriz and Thomas showed that every (finite or infinite) graph of tree-width k ∈ 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.
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Albrechtsen et al. (2024) studied this question.