Skunkvine, or Skunk Vine, "Paederia foetida" is listed as a noxious invasive that can regrow from bits of stem left in contact with the ground as well as the root system. The most environmentally friendly method of control is continuous cutting of any above ground growth to deprive, and eventually starve, the roots. There are some materials that, with repeated spraying apparently over several years, will eventually kill it. Spray environmentally hazardous poisons on the stuff for a long time and at great expense or keep cutting back the growth which will kill it probably in about the same time frame.

I have been pulling this vine up right at ground level and pulling up some of the root with it until it eventually breaks off for at least 10 years now but it still manages to come back. Right now, all of the shrubs are gone so I can keep an eye on it until I replant and resod in the spring. I don't want to use chemicals that will damage whatever I decide to replant there. I read where some have used an herbicide Garlon 3A or Plateau, however these are commercial products that were used on road side infestations and are probably too much for a residential application. At this point, it seems that my only options are to destroy the area with harsh poisons, remove all of the affected soil, or just buy another house.

Might there be someplace relatively nearby where this could grow and produce seeds and what you are getting is from germinated seeds?
Often an invasive species cna be very difficult to eliminate. We cut out a "Campsis radicans", Trumpet Vine, about 10 years ago and we are still pulling new growth from the roots of that vine. It does seem that we are seeing fewer as the years go by, however.