The crazy thing about the definition of NP-completeness is that Cook's theorem says that all problems in NP can be reduced in polynomial time to an NP-complete problem. So if a witness to a problem can be verified in polynomial time, it is by definition in NP and can be reduced to an NP-complete problem.

If I can verify a solution to this problem by finding a path in polynomial time, it is by definition in NP. The goal here was to present an example of a problem known to not be in NP.

> The crazy thing about the definition of NP-completeness is that Cook's theorem says that all problems in NP can be reduced in polynomial time to an NP-complete problem.

What were you trying to say here? Cook's theorem says that SAT is NP-complete. "All problems in NP can be reduced in polynomial time to an NP-complete problem" is just a part of the definition of NP-completeness.