Answer: The simplest way to deal with these persistent grapes is to cover your garden with black plastic mulch, and poke holes in which to plant your crops. You'll either need to put irrigation hose under the mulch or leave large holes in the plastic around the plants so moisture can reach the soil. This will at least limit the area in which the vines can sprout. Another option is a flame weeder. You don't actually set the plant afire, but the intense heat kills the green tissue. You'd have to use it regularly, since the roots will likely resprout, but eventually you'll exhaust their resources to produce more leaves. Hope these suggestions are helpful!
Also, are you aware that black walnut trees contain a toxin that prevents the growth of many other plants? Usually the toxin is leached from the soil a year after a tree has been removed from an area.
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