Question
What is the best lawn (NC) that can withstand a running Dobe.  She's an indoor pup so when we go outside, she will run up to 20 laps around the yard.  The area to the left of the driveway is mostly dirt now.  I had put down centipede twice but it can't stand up to her thundering feet!  Also, how to figure out why the grass under the grape myrtle is almost gone.  Ha ha ha  Image verification had r e y d (just need an "a" instead of the e and it would be yard...LOL)

Answer
 To answer your first question, "I have not uncovered the answer ...yet!" But here is how I learned to live with the problem. I used to 'stake out' a 20 ft. length of aluminum cable and clip Cody's leash to it. That way, he could run up and down. I would change the location of the stake-out weekly. The back yard could not win any landscape awards; but I never had to re-sod.

 Grass does not grow under busy feet (See question 1); nor under trees (look at the forest floor). If the sight of bare ground hurts your eyes, lay a thin layer of an organic mulch around the tree; rake / renew this once a year and it will do the 'crepe' or 'crape' myrtle, a lot of good. You might derive some spin-off benefit also. To maximize the latter, hand pull weeds out of the mulch. That will eliminate one cause of any problem you might run into with the tree itself.(No weed killer in the drip circle).
 Oh! I almost forgot, keep the mulch at least 6 inches away from the trunk; and do not run a weed eater anywhere near there.
 Answering this has been enjoyable; I trust that you will find it helpful.