How to prove NP-hardness of a Hamiltonian Path problem by reducing longest-path problem?

Question

I know how to prove longest-path problem by reducing Hamiltonian Path problem.

Here I want to prove NP-hardness of a Hamiltonion Path problem by reducing longest-path problem. (pretend we know longest-path problem is NP-hardness, not Hamiltonion Path problem)

I'm wondering if it is possible to reduce Hamiltonian Path problem to longest path problem.

Is there any way I can do that?

Answer 1

Suppose you have a graph G$G$ of order n$n$. G$G$ has a simple path of length ≥n−1$\ge n-1$ if and only if G$G$ has a hamiltonian path.