I am trying to understand the relationships between P, NP, NP-Complete and NP-Hard.
I believe I am starting to understand the general idea but, I am hung up on this question(see title).
What is an example of a problem that is not solvable in P time, is verifiable in P time but is not NP-Complete?
If there is some piece of understanding I am missing please fill me in.
Thanks in advance
As noted in the comments, this is the wrong site for this question. However, it can be answered briefly:
What is an example of a problem that is not solvable in P time, is verifiable in P time but is not NP-Complete?
If I understand you, what you want are problems that are (1) not in P, (2) in NP, and (3) not in NPC. Such problems are the NP-intermediate (NPI) problems.
It is not known if there is any such problem, because it is not known if P=NP.
If P=NP then clearly there are no such problems; if P=NP then also P=NPC, and therefore every problem which can be verified in P time is in all of P, NP and NPC because they are equal.
If P!=NP then it is known that there are such problems; at least one exists. Unfortunately we do not know if any real-world problems we face are in NPI provided that P!=NP. A list of likely candidates can be found here:
https://en.wikipedia.org/wiki/NP-intermediate
In short: knowing whether NPI is empty or not is equivalent to solving proving P!=NP, so get cracking! If you can find a problem that is definitely in NP but definitely not in P or NPC, then there's a big money prize awaiting you.