Memoirs of Yorkies Past: Beginning Breeders of Yorkshire Terriers.
I see you already have gotten expert advice, here's a link explaining Clarkwyn breeding (its just a so called original line) I know nothing of the breeder who's site the information is shared on not endorsing them.
Short answer is if your name is Smith, you have your line of yorkies you created after years of carefilly matched and championed breedings you'd probably name it 'or have itto called Smith line yorkies...or if you named york kennel Super Candy Awesome Fun Time (lol) it would be Super Awesome Candy Fun Time Yorkie Line...this naming is based on AKC and breed club line, kennel dog registrations ...maybe look up a sample 12 geberation pedigree on the net or someone can post one to show you.
My dog's name is Elvis...his AKC name is Castonague Shelbornes's Life of the Part Boy.