I do not have enough complexity theory background, but I was wondering about the kind of reductions that we normally do to show NP-Completeness. I think all of the reductions that I have seen are one-to-one; in that, given an instance $I_1$ of problem $P_1$, they map it to instance $I'_1$ of problem $P'_1$ such that $I'_1$ is a yes instance of $P'_1$ iff $I_1$ is a yes instance of $P_1$.
But I have never seen any reduction which can, say map an instance $I_1$ of $P_1$ to say $2$ or more instances of $P_2$ (in some sense, the mapping should involve creating some structure in $I_2$ where we have $2$ or more choices and we could pick any one arbitrarily). This will be kind of surprising (to me) if it exists, but nevertheless the following question bothers me even more.
Leaving that aside, I have never seen any reduction which might map $2$ or more (say YES instances) of a problem $P$ to the same instance of some problem $Q$.
I would appreciate if you can exhibit these reductions for NP-Complete problems.
NOTE: I understand that this is more of a cstheory question. But since this is certainly not research level, I decided to post it here.