Literature for restrictions that make NPC-Problems to P
While there are numerous individual examples of sharp thresholds (2-coloring vs 3-coloring is another one), probably the best structural result along the lines you're looking for is Schaefer's dichotomy theorem. 

Roughly speaking, the theorem looks at very general classes of problems (called constraint satisfaction problems) and completely characterizes (merely in terms of the kind of input) when the problem is NP-complete and when it's in P. 

I haven't read it, but the linked wikipedia page indicates a recent survey by Hubie Chen that presents this result in a more general framework.