Edge-ends versus topological ends of graphs

TL;DR

This paper characterizes when the topological ends of infinite graphs can be embedded into the space of edge-ends, revealing a combinatorial condition and identifying which edge-ends correspond to topological ends.

Contribution

It provides a purely combinatorial characterization of graphs where topological ends embed into edge-ends, extending Diestel and K"uhn's result to edge-end spaces.

Findings

Characterization of graphs with injective topological end embeddings into edge-ends

Identification of edge-ends that correspond to topological ends as those containing non-dominated rays

Establishment of a parallel between topological and edge-end spaces for certain graphs

Abstract

Diestel and K\"uhn proved that the topological ends of an infinite graph are precisely its undominated graph ends, yielding a canonical embedding of the space of topological ends into the space of graph ends. For edge-ends, introduced by Hahn, Laviolette and \v{S}ir\'a\v{n}, such an embedding does not exist in general. In this note, we characterize the class of infinite graphs for which the topological ends admit a natural injective map into the space of edge-ends that is compatible with the canonical maps between end spaces. Our characterization is purely combinatorial and is expressed in terms of edge-equivalence classes of vertices. Moreover, when such an embedding exists, we identify precisely which edge-ends arise from topological ends, showing that they are exactly the edge-ends containing a non-dominated ray. This establishes a parallel result to the theorem of Diestel and K\"uhn for edge-end spaces.