You do not specify, but unless the section of yard you are considering as a planting site is difficult to mow due to slope, here is a somewhat nontraditional suggestion: Would you consider creating a sort of canine playground?
Rather than plant grass that may not grow or plants that will be unhealthy and not thrive, why not let it all go to the dogs? (Pun intended.) Border the area off in a material to your liking (stone, brick, wood), and prepare it as you would a sandbox. Line the bottom and sides with water-permeable landscaping material to prevent weed growth from underneath, and fill the area with approximately 8 inches of shredded bark, pine needles, or even pea gravel. The dogs will enjoy digging a bit for a cool nest without reprimand, and the area will be easily cleaned.
If you prefer, it may be possible to maintain plants on the site, but to determine which type is suitable will require further consideration. Soil compaction caused by heavy traffic is really the main obstacle to plant health in this situation, other than urine burning, of course.
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