Efficiently distinguishing all tangles in locally finite graphs
arXiv:2303.09332 · doi:10.1016/j.jctb.2024.03.004
Abstract
While finite graphs have tree-decompositions that efficiently distinguish all their tangles, locally finite graphs with thick ends need not have such tree-decompositions. We show that every locally finite graph without thick ends admits such a tree-decomposition, in fact a canonical one. Our proof exhibits a thick end at any obstruction to the existence of such tree-decompositions and builds on new methods for the analysis of the limit behaviour of strictly increasing sequences of separations.
19 pages, new revised version: fixed several small typos, included appendix into main body of the paper, added new proof of Lemma 3.10, added two figures and fixed proof of Lemma 4.5