Characterizing the Infinitely Visited Intersections in a Ride-Forever Path on a Directed Graph

A neighborhood is a set of one-way and two-way streets connected by intersections. Let our home be the starting point of an infinite route. This could also be seen as a directed graph $G= (V, E)$ with a start vertex s. Let a ride-forever $p$ of $G$ be an infinite sequence of intersections (or vertices), beginning at s and never-ending. Clearly, since $V$ is finite, there must be some vertices that are visited an infinite number of times in $p$.

If a neighborhood has a ride-forever, let $RF(p) \subset V$ be the set of intersections that occur infinitely often in $p$. Explain/Show why $RF(p)$ is a subset of a single strongly connected component of $G$.


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