Answer: Digging is the most permanent solution, and it takes persistence. You might elect to use an herbicide such as a product containing glyphosate to control the vines. The best time to apply this type of herbicide is when the plant is actively growing. The reason for this is that the plant translocates the chemical from the leaf to the root where it kills it.
You must be patient while this process occurs because the leaves will brown and look dead long before the entire process is complete. Be sure to follow the label instructions. Apply it very carefully when working near desirable plants because this is a non-selective herbicide and as such it will be absorbed by all green leaves it touches.
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