A graph end is an equivalence class of one-way infinite graph paths in an infinite graph. Two such paths represent the same end iff, after deletion of any finite vertex set, tails of both lie in the same connected component (Aurichi et al. 2026). A tail is the part of a graph path remaining after removal of a finite initial segment. A one-way infinite path graph has one end, and a two-way infinite path graph has two.
Write 
for the set
of graph ends of a graph 
. For a locally finite graph,
the space 
is obtained by treating each graph edge as a unit
interval and adjoining 
.
A basic neighborhood of an end consists of the connected component containing its tails after
deletion of a finite vertex set, all ends with tails
in that component, and adjacent half-open edge intervals. In this space, the set
closure of an acyclic subgraph
may be homeomorphic to a circle.
This distinction underlies the topological variant of linear
arboricity studied in connection with the linear
arboricity conjecture.
