On the Fastest Moving Off from a Vertex in Directed Regular Graphs

Let $\Gamma$ be a directed regular locally finite graph, and let $\overline\Gamma$ be the undirected graph obtained by forgetting the orientation of $\Gamma$. Let $x$ be a vertex of $\Gamma$ and let $n$ be a nonnegative integer. We study the length of the shortest directed path in $\Gamma$ starting at $x$ and ending outside of the ball of radius $n$ centered at $x$ in $\overline\Gamma$.