Ask Mr. Smarty Plants

There are 10 species of Wisteria, of which two are native to North America and the other eight are from Asia. It is possible that the plant that is invading your garden is Wisteria frutescens (American wisteria) which is native to Georgia. However, it is much more likely it is either Wisteria sinensis (Chinese wisteria) or  Wisteria floribunda (Japanese wisteria). The Asian species are the ones most often found in home gardens, and are much more invasive than the natives. It really doesn't matter too much if you have the native or non-native plants, they are obviously invasive in your garden. 

This Plant Conservation Alliance Alien Plant Working Group article Least Wanted-Exotic Wisteria has some information and good suggestions for treating this problem. You are in a situation where you can't fight the problem at its source, and must deal with it after it gets on your property. This is going to require frequent attention, and not letting it get ahead of you. The very first thing you need to do is go out and cut the vine going up your oak tree off at the root and then cut the girdling vines at intervals up as far as you can get. Any time you see a sprout of wisteria coming up, don't just snip it off, try to dig it out. Where the vines come over your fence, cut them off, and paint the cut surface (right away) with a 25% solution of glyphosate herbicide with water. Don't spray it, that will only take out desirable plants around the wisteria. Vines that are coming up on your property should be cut off at the root collar and, again, quickly painted with the herbicide. Be very careful not to let the herbicide run off into the soil, and keep it out of contact with any portion of a desirable plant. A disposable paintbrush is about the best means of applying the herbicide to cut surfaces, but it needs to be done in about the first 5 minutes after cutting, because the wound will seal over. 

As long as the "mother" plant is still alive, there will continue to be incursions into your garden. Don't allow yourself to get lax, or think you have the problem solved because you've treated everything. It will be back!