How can I get rid of "Green Carpet" (Japanese Spurge)?
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Short of digging the whole patch up, what is a pet safe way to get rid of Japanese Spurge?
what else is in the garden
Contact a local garden center. They may be interested in the pachysandra to re-sell. It is a great ground cover. The variegated Bishop's Weed is actually the problem, as it propagates vegetatively (by cuttings). That may take longer to get rid of.
Weed killer: 1 gallon vinegar and 1 Tables dawn liquid soap. Some say to avoid putting 1 cup of epsom salt in or it kills everything.
Here's a post that might help https://chascrazycreations.com/home-made-weed-killers/
Try to tackle your purslane problem while the plants are young to avoid scattering seeds. If you are hand-pulling mature plants, cover the cleared area with a black plastic gardening tarp or heavy-duty trash bags for four to six weeks. The sunlight on the black plastic heats the topsoil to kill any residual seeds and blocks sunlight from any purslane fragments that attempt to sprout again.