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.
For a locally finite graph 
, the space 
is obtained by treating each graph
edge as a unit interval and adjoining its set
of ends. 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.
