Showing NP-Completeness

I just newly started looking into computational complexity.

Since we don’t know if P = NP, we would like to have a way of saying “This problem is in NP and is really hard unless P = NP.”

This is made more formal by the notion of NP-completeness.

Therefore, we say a problem A is NP-complete if

(a) it is in NP, and (b) every problem B in NP is polynomial-time reducible to A.

Using a graph $$G$$ and an edge $$e$$ in $$G$$ as an example, how do I determine whether or not there is a Hamiltonian cycle that includes $$e$$?

With this, I could show that the problem is NP-complete and use the same approach for similar problems.

There is no known efficient way to determine whether or not there is a Hamiltonian cycle that includes $$e$$. That is a NP-complete problem. (If you could determine that, you could use that as a subroutine to test whether the graph contains any Hamiltonian cycle, and even to find a Hamiltonian cycle if one exists; so it's not reasonable to expect there to be an efficient algorithm to do that.)