Yes. There are posted signs for pet friendly monuments. Actually there were a lot of dogs out throughout the city.
I did not see anyone with pets when I was there. However that doesn't mean their not allowed. You can probably find out for sure by checking with the Martin Luther King Jr store that's at the memorial and they could tell you for sure.
I did not see any pets. But I'm sure if you clean up after them you should be good to go. Monument area was concrete. Grass not to far off.
I did not see anyone walking a dog at all so I'm not sure but would assume so since it's an outside public place
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