Answer: I think the plant you have is Hedera helix or English ivy. This plant is sometimes used as a groundcover in areas where lawn grass will not grow or where it is difficult to mow -- such as shady slopes or in competition with surface tree roots for example -- so be sure you really do want to eradicate it. It will probably take several years of effort before you succeed. Cutting will not work as it returns from its underground root reserves. You need to dig out or otherwise kill the roots.
English ivy is very persistent so in most cases it is necessary to use an herbicide such as triclopyr or glyphosate to eradicate this plant. The triclopyr will not kill grass, but the glyphosate is nonselective and will kill all plants it contacts. These are absorbed through the foliage and stems and translocated to the roots. In addition to spraying the foliage, they can be wiped or painted onto the cut stems to reduce the amount being used.
The following invasive plant information developed by the National Park Service and invasive plant experts includes instructions for optimal use of the chemicals as well as methods of digging out the vines. Also be sure to read and follow all of the label directions on whatever product you use for best results.
Good luck -- this is a huge job.
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