The number of generations required varies from breed to breed, but all pedigreed animals have papers from the registering body that attest to their ancestry.
This is because current accepted breeding practices within the pedigreed community results in a reduction in genetic diversity, and can result in physical characteristics that lead to health issues.
Accidental crossbreeding by feral animals may result in harm to breeding programs of pedigreed animals; their presence may also excite domestic animals and push them to escape.