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arXiv · 2205.09865

End spaces and tree-decompositions

Marcel Koloschin·Thilo Krill·Max Pitz
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Abstract

We present a systematic investigation into how tree-decompositions 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 distinguished, and characterise which subsets of ends can be displayed by a tree-decomposition of finite adhesion. In particular, we show that a subset $Ψ$ of the ends of a graph $G$ can be displayed by a tree-decomposition of finite adhesion if and only if $Ψ$ is $G_δ$ (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 $G_δ$, this provides a structural explanation for Carmesin's result that the set of undominated ends can always be displayed.

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Marcel Koloschin, Thilo Krill, Max Pitz. 2023-05-16. End spaces and tree-decompositions. https://arxiv.org/abs/2205.09865

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