Find the vines during fall and winter months, when the weed is lying dormant. Look for vines beneath trees, bushes, downward yard slopes and ditches. Notice dark brown or green heart-shaped leaves. Measure the leaves to ensure you don't remove the traditional, and unharmful, yard vine morning glory. If the leaves stretch 8 inches in diameter, you have found air potato vines.
Snip the "potatoes." Place the blades on a pair of sharp hand-held garden shears around the potato stem. Clip off all potatoes you find and dispose of them in the garbage. Identify potatoes by their tan or brown coloring and oval shape. Potatoes will rest along the ground with the vines.
Spray the vines with herbicide. Purchase an herbicide with triclopyr. Triclopyr is a natural weed killer, mixed with elements of ethanol and kerosene. Follow herbicide labels for mixing instructions. Pour the herbicide into a spray bottle. Saturate the vines directly with herbicide. While triclopyr does not typically harm grass, it does not always discriminate and can kill nearby plants, so read all mixing and application directions and use wisely.