They sell dog repellant sprays at pet stores and sometimes hardware stores, too, and you could try spraying a line of that along the border where he crosses over. You could also get yourself a big container of cayenne pepper and sprinkle that generously in the areas the neighbor dog tends to frequent. That'll burn his nose and make him sneeze and retreat. A couple of huffs of cayenne ought to teach him a lesson. There are probably other odors that are natural dog repellants, but I don't know what they are. Do some research on Google.

The person who needs to be put to sleep is actually your neighbor herself. That's really creepy that she lets her dog crap in your yard.