Sandra Albrechtsen, Raphael W. Jacobs, Paul Knappe, Max Pitz
Published 2024-05-10Version 1
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.
Connected even factors in the square of essentially 2-edge connected graphs
Graphs with large generalized 3-connectivity
On (strong) proper vertex-connection of graphs
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