A metrization theorem for edge-end spaces of infinite graphs
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911070475517952 |
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| author | Pitz, Max |
| author_facet | Pitz, Max |
| contents | We prove that the edge-end space of an infinite graph is metrizable if and only if it is first-countable. This strengthens a recent result by Aurichi, Magalhaes Jr.\ and Real (2024).
Our central graph-theoretic tool is the use of tree-cut decompositions, introduced by Wollan (2015) as a variation of tree decompositions that is based on edge cuts instead of vertex separations. In particular, we give a new, elementary proof for Kurkofka's result (2022) that every infinite graph has a tree-cut decomposition of finite adhesion into its $ω$-edge blocks. Along the way, we also give a new, short proof for a classic result by Halin (1984) on $K_{k,κ}$-subdivisions in $k$-connected graphs, making this paper self-contained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_16625 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A metrization theorem for edge-end spaces of infinite graphs Pitz, Max Combinatorics General Topology 54E35, 05C63, 05C40 We prove that the edge-end space of an infinite graph is metrizable if and only if it is first-countable. This strengthens a recent result by Aurichi, Magalhaes Jr.\ and Real (2024). Our central graph-theoretic tool is the use of tree-cut decompositions, introduced by Wollan (2015) as a variation of tree decompositions that is based on edge cuts instead of vertex separations. In particular, we give a new, elementary proof for Kurkofka's result (2022) that every infinite graph has a tree-cut decomposition of finite adhesion into its $ω$-edge blocks. Along the way, we also give a new, short proof for a classic result by Halin (1984) on $K_{k,κ}$-subdivisions in $k$-connected graphs, making this paper self-contained. |
| title | A metrization theorem for edge-end spaces of infinite graphs |
| topic | Combinatorics General Topology 54E35, 05C63, 05C40 |
| url | https://arxiv.org/abs/2507.16625 |