Indeed, there is probably nothing more effective at killing rats than "ratter" breeds of smallish dogs. These would include terriers and some dachshunds (the more aggressive and active ones). Look for a standard size Dachshund who is lean, alert, and active, and you have an excellent ratting dog.

The rat will either be exterminated or he will depart the area. This way the rat has a good chance to make up his own mind about leaving, and I like that approach.