Clemson.edu info here. All below was extracted from that page.
A third common greenbrier in SC is Smilax glabra (called the cat greenbrier), which can be distinguished from other species by its pale-colored, lower leaf surface. The leaves are typically mottled (that is, lightly spotted) but have no spines along the leaf margins.
Chemical control of greenbriers is difficult because their extensive root system can regenerate new vines from further back along the knobby rhizomes, and the waxy foliage resists the uptake of sprays. If the greenbriers to be controlled are only a few small individual plants, it is possible to dig up the rhizomes. However, if it is a larger vine, then chemical treatments will be necessary.
Because most chemical sprays may not penetrate the waxy coating on mature foliage, cut the vines and spray after they re-sprout tender new growth. Wait until the regrowth is ½ to 1 foot tall and spray with a 10% solution of glyphosate. To make a 10% solution, add 13 fluid ounces of a 41% concentrate glyphosate product with enough water in a pump-up sprayer to make a gallon of spray. Alternatively, wait until early spring to spray new growth. See Table 1 for examples of products containing glyphosate.
Beneath desirable shrubbery, cut the vines near the soil line and pull out the vines. Immediately paint or spray the freshly cut vine stumps with a 10% glyphosate solution, but do not allow the herbicide to touch landscape plants. Glyphosate has very little soil activity and should not be absorbed by the roots of nearby landscape plants.
Triclopyr is a broadleaf herbicide that is absorbed by the mature foliage of greenbrier vines. Spray the foliage with a solution of triclopyr (9 fluid ounces of a 61.6% product with water to make a gallon of spray, or a 50:50 mix of an 8 or 8.8% product with an equal amount of water). See Table 1 for examples of products containing triclopyr.
Alternatively, spray or brush the triclopyr solution onto the freshly cut stumps of greenbrier vines for control. There is some soil activity with triclopyr, so do not use products containing triclopyr near desirable landscape plants, and do not allow the triclopyr solution to contact the trunks, stems, or foliage of desirable plants.