Uh oh.
Yorkies should have an IV during surgery for fluid and an emergency line to them. They should not use IV anesthesia to maintain them. You need to use a gas anesthesia for Yorkies (all dogs actually), either Isoflourine or Sevoflourine. This is given through a tube down the throat. Some vets give this gas, before they put the tube in the throat, through a mask, so they can place the tube. Other vets do use an injectable anesthetic only so they can intubate. These would be extremely short acting. Either way is acceptable but Yorkies should never go through an actual surgery without gas anesthesia. You also have to be careful that they don't use barbiturates as the injectable before the gas. I believe one exception to this would be Propofol which is very safe. I would be talking to your vet...