No. A wire doesn't have to form a complete loop to work as an electric dog fence. The wire just has to be in contact with the ground (or water) so that electricity can flow through it. This is why you'll often see electric dog fences with a wire running along the edge of the property, rather than forming a complete loop around it.

Yes. If you have an electric dog fence, it needs to create a complete loop around your property in order to keep your dog contained. If there is an opening in the fence, your dog will be able to escape and could get injured or lost. Make sure that the fence is secure and that there are no openings before you let your dog roam freely in the yard.