I did a bit of searching regarding my previous post, and it is known that if P != NP then there do exist
NP Incomplete problems. They're actually called "NP Intermediate" rather than "NP Incomplete" problems. I like my name "NP Incomplete" better but the term "complete" may have a specific meaning that doesn't apply to the NP intermediate class --- I don't know.
The existence of NP incomplete problems was proved (assuming P != NP) by Richard Ladner in 1975. I haven't looked at it but hope to soon. Some well known problems thought to be NP Intermediate are:
- Graph Isomorphism
- Factoring
Garey and Johnson also mentions Linear Programming as a candidate for NP intermediate, but we now know that LP's are in P. It's good to remember we are making progress.