Answer: Unfortunately, based on your description I am not certain what this plant is. However, there are three approaches to dealing with the encroachment. One would be to talk to your neighbor and see if the vines can be controlled at the source. This would probably be the most effective. The next is to routinely cut the vines at the fence line before they spread further into your property. The third and least desirable approach would be to investigate the local weed ordinances.
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