Question 158
Let $G$ be an infinite directed graph, every vertex of which has finite degree. Suppose that, for every vertex $s$, the in-degree of $s$ is strictly less than the out-degree of $s$. Let $v$ be a vertex of $G$. For every integer $n \geqslant 1$, let $V_{n}$ denote the number of vertices that can be reached from $v$ by passing through at most $n$ edges, $v$ included.
What is the smallest possible value of $V_{n}$?
What is the smallest possible value of $V_{n}$?