Answer: There are several plants you could use. English ivy (Hedera helix) and periwinkle (Vinca minor) are evergreen groundcovers that do well in shade or morning-only sun or bright dappled light all day. Ajuga is semi evergreen and will do well in shade or sun. It can range in height from extremely short to about six inches tall depending on the variety you select. These should be available as very small starter plants, set them about a foot apart (or closer for faster coverage.) Be sure to mulch in between the plants with an organic mulch to help keep down weeds and reduce watering needs while they become established. I should mention that if the dog ruined lawn by running, then the groundcover may also suffer wear. You might consider using a large particle mulch or possibly gravel in that area rather than plants.
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