Image of a graph under a covering map is a graph

Question

It is well known that a covering space of a graph is also a graph. It is also true that the image of a graph under a covering map is a graph? I'm not sure that this is true, but I cannot find a counterexample.

Answer 1

This is true. Topologically a graph is a topological space which is locally homeomorphic either to an open interval in $\mathbb{R}$ or to a "star" (a point with some number of open intervals attached to it; a typical small open neighborhood of a vertex in a graph). Since covering maps are local homeomorphisms, if $f : X \to Y$ is a covering map and $X$ is locally homeomorphic to the above spaces then $Y$ must have the same property. Said another way, if a point $y \in Y$ has no neighborhoods homeomorphic either to an open interval or to a star, then the preimages $f^{-1}(y) \in X$ also cannot have this property.