A metrization theorem for edge-end spaces of infinite graphs

TL;DR

This paper characterizes when the edge-end space of an infinite graph is metrizable, showing it is so if and only if it is first-countable, and introduces new elementary proofs for related graph-theoretic results.

Contribution

It provides a new characterization of metrizability for edge-end spaces and offers elementary proofs for key results on tree-cut decompositions and graph subdivisions.

Findings

Edge-end space is metrizable iff first-countable.

New elementary proof for Kurkofka's tree-cut decomposition result.

Short proof for Halin's result on $K_{k, ext{kappa}}$-subdivisions.

Abstract

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 $\omega$-edge blocks. Along the way, we also give a new, short proof for a classic result by Halin (1984) on $K_{k,\kappa}$-subdivisions in $k$-connected graphs, making this paper self-contained.