plant called mind your own businness

I have mind your own business which I planted between my path and the fence, as there was an untidy looking narrow strip, I had put yellow pebble stones there but they kept getting skitted about by the dog overtaking me on the path!!!! so I planted Mind your own business which has made a nice green border and does not seem to mind the four legged trampling! As far as it is spreading into the border I just pull it up and dispose of it. I have not had the problem in the lawn. I would imagin a spot weedkiller on the patches. Not knowing to what extent it has spread. As far as the cause I can only think that a piece has been dropped there and trodden in as it will spread as you have discovered..

The smallest piece of this will root and regrow - I've been trying to get rid of it for a long time :-(

I've got in in my flower borders. I haven't managed to completely eradicate it when I tried. In the end I decided to just live with it and pretend it's deliberately there as a groundcover... It's not in my lawn though as a path is in the way.