End spaces and tree-decompositions

We present a systematic investigation into how tree -decom-positions of finite adhesion capture topological properties of the space formed by a graph together with its ends. As main results, we characterise when the ends of a graph can be dis-tinguished, and characterise which subsets of ends can be displayed by a tree-decomposition of finite adhesion. In particular, we show that a subset psi of the ends of a graph G can be displayed by a tree-decomposition of finite adhesion if and only if psi is G delta (a countable intersection of open sets) in |G|, the topological space formed by a graph together with its ends. Since the undominated ends of a graph are easily seen to be G delta, this provides a structural explanation for Carmesin's result that the set of undominated ends can always be dis-played.(c) 2023 Elsevier Inc. All rights reserved.