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, <monospace>arXiv:2404.17116</monospace> , 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi> ω </mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Subscript k comma kappa"> <mml:semantics> <mml:msub> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi> κ </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">K_{k,\kappa }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -subdivisions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -connected graphs, making this paper self-contained.