The one with runners sounds like nutsedge. Nutsedge likes moist soil so if you are watering this area cut off the supply. And since it has infested itself, the only thing you can do is to keep handpulling on a continous basis - the other way it spreads is by self-seeding.

I saw a product called Sedgehammer that intrigued me, but I haven't tried it. I had the same problem when I first took over a property a couple years ago: I hand pulled it every week during the season and didn't let it go to seed. Barely an issue anymore. Quite the pain when it's amongst other bladed plants! My commiserations.

Sedgehammer is labeled for turf and woody ornamentals. It would probably damage the Liriope. I have used it in turf and it works really well. They also have a supplemental label for Horsetail control.
I believe that yellow nutsedge has sterile flowers so it won't reseed. It does have nutlets which are like little bulbs and are incredibly persistent. Check out the next plant you pull and you should see the nutlets forming on the tips of the roots.

You do not have to broadcast a herbicide. They can be applied with an artist brush in tight place like flower beds.

Dig it all up, separate out the weeds and replant. You may have to replace the soil first so that there is not re-infestation from remaining fragments.

I have used manage min. concentration. the liriope looked a litte sad for a week or to but did not die. maybe I just got lucky.