Is it compulsory that the 'reduction of p‌r‌o‌b‌l‌e‌m be done in polynomial time' for it to be NP complete?

Question

For a problem to be NP complete, it must belong to the class NP and there must be a polynomial time algorithm to reduce it to an NP complete problem .

Now what if we only have an exponential time algorithm for reduction . Will this problem still be called NP-complete ? Or are there no such existing problems?

EDIT : Also please tell me whether there is any such problem and if it exists then to which class does it belong ?

Answer 1

It can only be consider NP-complete if other NP problems can be reduced to it in polynomial time. The reason this is a useful definition is that if we find a polynomial time algorithm for one, it automatically gives one for all NP problems. If we allowed an exponential time reduction, but found a polynomial time solution to the reduced problem, that wouldn't actually help us solve the one we reduced it to.

Hope this helps.