Problem: Longest Path
Input: An undirected graph $G=(V,E)$
Question: Is there a path of length at least $\frac{|V|}4$?
I know that in order to prove the simple version of $k$ longest path, we reduce the Hamiltonian path to longest path. How we can prove the $NP$-completeness in this special case? Should I use a reduction from Hamiltonian cycle?
Edit: A hint : Reduction from Hamiltonian path. Try to think how many vertices should be added to a special instance $G'$ of Longest path problem so that if there is a path least $\frac{|V|}{4}$ vertices in length then there is a Hamiltonian path in $G$.