Harder to solve, not necessarily harder to verify?

If I am understanding things right.

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.