Hello. I had this problem a few years back and know exactly the situation you are in. The ones that I could not get rid of by digging up I cut them as close as possible to the ground,drilled a hole in the middle of the stem and poured brush killer(Bayer)in the hole. If your leary of using commercial weed killers try straight vinegar and salt. Good luck.

Vinegar and salt won't phase wisteria.

You can use herbicides in a safe manner.

Use one with the active ingredient triclopyr according to "paint on" directions wherein you cut the stem near the ground, then immediately paint the cut surface of the stump with the weed killer.

Success will require persistence and multiple treatments. So, repeat as needed.

Wow, there's a weed I wouldn't mind being overrun with! Doesn't grow where I live unfortunately.

Wisteria can be one of the most beautiful vines in the world when properly managed. By the sound of it yours has been abandoned.

Rather than kill it, what about hiring a specialist to come in and prune it and teach you how to look after it.

Try doing a search for a Master Gardener group in your area. If you let us know where you live I will try and find some resources for you.