Does Covering Map of Graphs map trees to trees?

Question

Let $f: C\to G$ be a covering map of graphs (https://en.wikipedia.org/wiki/Covering_graph).

If $T$ is a tree in $C$, is $f(T)$ a tree in $G$?

Thanks.

Answer 1

I presume you meant to ask if $f(T)$ is a tree in $G$? The answer is no. For example, $G$ could be the figure 8 graph, and $f : C \to G$ could be its universal covering map, so $T=C$ is a tree and $f(T)=G$ is not a tree. In fact, you could even take $T$ to be any single edge of $C$ and then $f(T)$ would be a loop in $G$ hence not a tree.