The Rho Shape

Start at any node and keep following edges. You visit $x$, then $f(x)$, then $f^2(x)$, and so on. Since you take one step per iteration, you are building a sequence. Since the graph is finite, you must revisit some node. Once you do, you have found a cycle.

You cannot escape because $f$ is deterministic. Every path looks like a tail leading into a cycle. The shape resembles the Greek letter rho: a straight line that curves into a loop. This structure is universal in functional graphs.