The AKC registration does not prevent inbreedings. You can see by the pedigree of Ch. Jubilee Eagle (
http://tx2stepnyorkies.com/photo_7.html) that his sire was a mother/son breeding. There are breeders that do this type of breeding all the time and they are registered with the AKC.
This is what the AKC states on their website under FAQ:
"Why do you allow inbreeding and line breeding?
The relationship between the Sire and Dam of the litter has no bearing on registration eligibility. Anyone who considers breeding their dog must consider temperament, health, and finding mates that complement each other. Breeding dogs that are closely related can fix certain characteristics, both desirable and undesirable. Therefore, we would expect that only very experienced breeders would consider inbreeding and line breeding. However, there is no rule that prohibits inbreeding and line breeding; this is left to the discretion of individual breeders.
Inbreeding - The mating of two closely related dogs of the same breed.
Line breeding - The practice of mating a dog to a member of an earlier generation of the dog's bloodline."
Just thought you should know.