A metrization theorem for edge-end spaces of infinite graphs

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 [Preprint, arXiv:2404.17116 , 2024]. Our central graph-theoretic tool is the use of tree-cut decompositions, introduced by Wollan [J. Comb. Theory Ser. B 110 (2015), 47–66] 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 [Math. Ann. 382 (2022), 1881–1900] that every infinite graph has a tree-cut decomposition of finite adhesion into its ω ω -edge blocks. Along the way, we also give a new, short proof for a classic result by Halin [Advances in Graph Theory, Annals of Discrete Mathematics, vol. 3. pp. 98–109, Elsevier, 1978] on K k , κ Kk,κ -subdivisions in k k -connected graphs, making this paper self-contained.