A graph cover, also called a graph covering, of a graph 
is a graph 
together with a graph projection 
such that the restriction
of 
to the edges incident to each vertex
of 
is a bijection onto the edges
incident to 
in 
(Gross and Tucker 1987). Equivalently,
the neighborhood of every lifted vertex 
in the cover looks locally like the neighborhood of its
image 
in the base graph, with incident edges corresponding
one-to-one.
A vertex 
of 
with 
is called a lift of 
.
A graph cover should not be confused with an edge cover, which is a set of edges whose endpoints contain all the vertices of a single graph.
A voltage graph gives a compact way to construct regular graph covers. A graph cover is a regular graph cover when its deck transformations act transitively on each fiber of the graph projection. Here "regular" refers to the covering action, not to a regular graph. More generally, permutation voltage assignments generate arbitrary graph covers (Gross and Tucker 1977).
