Let $G=(V,E)$ be a graph and consider a random walk on it. Let $G'=(V',E')$ be a subgraph consisting of the vertices and edges that are visited by the random walk.

Question 0: Is there a standard name for $G'$?

Intuitively $G'$ is a thin subgraph, so for instance, even when $G$ is transient, $G'$ can be recurrent.

Question 1: Is there a counterexample? So, Is there a transient graph $G$ so that $G'$ is transient with positive probability?

I'm also curious to know what happens when one iterates this procedure, $G,G',G'',\dots$. Does it eventually look like a path graph?

Question 2: What can one say about $G^{(n)}$ as $n\to \infty$?