As noted below this is a regular graph reachability problem with no real performance requirements, and there're already many such katas available.

For examples see the Example Test Cases

This is not the right way to explain a task.

The description obfuscates this, but it turns out this is just a generic graph search kata like countless before it.
If the point of it is to enforce solving with a specific, efficient algorithm, then the performance tests need to be much much stricter. As it stands, a standard BFS implementation solves this in ~5s.