I have been battling purple nutsedge (Cyperus rotundus) and yellow nutsedge (Cyperus esculentus) as long as I've had a lawn. These weeds grow from tubers. The tuber networks are connected by a baby hair fine root system--ordinarily too fragile and deep to extract without snapping. Whether just a myth or not, I'm told that each connection I break will eventually bring forth three more terrible tubers.
I live in the southwestern part of the United States. The year we re-sodded the front lawn, I spent hours over the summer digging for tubers anytime something vaguely appearing to be nutsedge sprouted. Every year since, though, the nasty buggers return.
I'm familiar with chemical approaches to selectively controlling nutsedge in a lawn (see table 1. Controlling Nutsedge with chemicals, here); hoping someone has discovered an organic solution. I'd prefer something that didn't require me to spend every summer camped on my front lawn with a bucket and spade.