DHAMPATH is the set of all the directed graphs which have a hamiltonian path. It's a well known NP-complete problem.
I know the proof of SAT's downward self-reducibility. Arora-Barak says Cook-Levin's proof shows that all NP-complete languages are downward self-reducible. How?
I feel DHAMPATH can directly be shown downward self-reducible without using Cook-Levin, but I am not able to. Can you give some hint?