I have a four foot pink Mimosa tree. It has root damage from shipping. I put it in a plant pot with good drainage and have watered the plant. It is winter so the tree is dormant as well.
A picture's worth a thousand words. If the damage was only minor (i.e. no bigger branches of the root have been bitten off by your dog) then leaving it as is (and not watering it while it's dormant, this also means having it under a roof to shelter it from heavy rains) will be enough. If the damage is quite extensive (i.e. bigger parts of the root have been bitten off) then you might want to try IBA. It's a powder that needs to be dissolved in an organic solvent (usually ethanol) first and then diluted by water to get the desired (ppm) concentration.