The main problem with dogs on lawns is that of the brown patches caused by bitches urinating. Male dogs are not such a problem in this respect, for they do it differently..
The brown patches tend to be a circular patch with bright green grass growing around the perimeter as the bitches urine eventually breaks down to Nitrogen, which then feeds the area outside the actual damage. The brown patch is basically scorching - similar to that caused by applying too much fertilizer - especially in dry conditions.
Treatment by soaking the area with water - immediately after the bitch has done the deed! Soaking the brown patches after they have formed will be absolutely essential if you are going to over-seed the patches.
As with many things, prevention is better than cure. You - the owner - have the remedy. Keep your bitch off the lawn!
There are several applications available at most garden centres to keep your dogs away from certain areas. However, if you allow your dog to use the lawn as a toilet, then these applications are of no use. If a dog has to go, a dog has to go!