If we prove that there is an NP-complete problem that is P, Can we consider that P=NP? I discover this in All NP problems reduce to NP-complete problems: so how can NP problems not be NP-complete?


If problem B is in P and A reduces to B, then problem A is in P.
Problem B is NP-complete if B is in NP and for every problem in A in NP, A reduces to B.
Problem C is NP-complete if C is in NP and for some NP-complete problem B, B reduces to C.


My questions are (if I then II then(?) III/I. If III/I and III/II then IV.)

I: Are there a generalized form to reduces NP problem to either P or NP-complete?
II: Are there a certain number of NP-complete problems?
III/I: Are all of the NP-complete problems can be reduces to all other NP-problems?
III/II: If we can reduce B in NP-complete problem to A in P, can we prove that all problem reduces to B is in P?
IV: If we prove that there is an NP-complete problem that is P, Can we consider that P=NP?
I. I was not able to parse the question, sorry.
II. There is an infinite number of NP-complete problems.
III/I. Each NP-complete problem can be reduced to any other NP-complete problem.
III/II. If you show that there exists a (Karp) reduction from a NP-complete $B$ problem to a problem in P, then you have proved that all problems that reduce to $B$ are also in P. In particular this implies NP $\subseteq$ P, and hence P=NP.
IV. If a NP-complete problem is in P, then P=NP.