On the fastest moving off from a vertex in directed regular graphs

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