Hi, that potato vine sounds interesting, and strange at the same time. My husband and I have a problem with Trumpet Vines and are going to be starting in a day or two to take them down and kill them. I know what you are going through...the feeling of loss when you try so hard. This is what I would do. The ones that are woody, dig out the roots and get every piece of the root and throw it in a bag, not a compost pile. Dispose of it. Then anything that shows any green, paint those leaves with Round Up so that it takes it down to the roots you may have missed. When chopping the roots, paint it on the newly cut root as well. I would paint the RoundUp on your leaves of all your vines, if you are sure you want everything gone. Be careful not to spray the Round Up as you may get it on other flowers, grass, etc. Wear safety glasses and gloves and beware of the wind/breeze. I hope this helps. Not sure what you have and it is too bad that we have to resort sometimes to chemicals which is the last resort...but sometimes must be done.

I posted this message on the Native Flowers board, but I also figured I'd post it here in case it's not a native. By the way, I'm in Northwest Ohio. I just cleaned up my fern bed and there was a woody vine or runner of some sort growing all over. It was anchored every few feet and several vines branched off at each anchor point. It had smallish, regular oval pointed leaves and small pink flowers (it had just started to flower a few days ago). The flowers looked almost honeysuckle-like but much smaller and had no noticeable scent. There are two very old yew bushes at the back of the bed and the vines did not grow up them at all, though at some points in the middle of the bed the vine grew upwards and branched off like a shrub. I don't know if it's a native plant or not but it showed up a couple years ago as a volunteer and really took off. I don't have a means to post pictures so I hope this description is enough. Any help/pictures of what you think it might be is appreciated. Thanks!

I finally did something with the strip between my yard and the farmer's field in back. There is all that you and marcia_m mention. They can and will kill trees. Last year I started out with loppers, then my lawnmower but gave up; then my step brother decided to bring his brush mower over, only it's not a farm tractor attachment, it's a mower from Sears with weedeater type line. He swore it would go through that mess "like butter". HA. I knew better. The lines kept breaking over and over. It did get alot of the weeds but not the multi flora rose and woody vines with their big trunks. And the thorny stems kept whipping around and grabbing at his bald head. LOL. Finally he looked at me with sweat dripping off him and said, "I'm going home to get my chain saw". Even with that it was a chore, but we dragged four or five huge MR shrubs out of there along with the myriad of Va.creeper and wild grape. That was in the Fall, and I followed up by spraying brush killer on the exposed stumps of the woody plants. That helped. This Spring I'm keeping the area mowed down with my riding mower and still spraying the leaves of the noxious vines. I don't think I will get rid of them completely - just keep them under control.

Following up with the final determination, some good news and some bad: Starting with the good: There is in fact a simple and relatively foolproof way to ID the natives in my area and distinguish them from the invaders! The bad? I've got me several acres of heartache and pruning. I reached out to a botanist at the New England Wild Flower Society, in hopes of clarifying the 'should it stay or should it go' dilemma. After a brief scolding about how invasives shouldn't be ostracized, they didn't choose to come here, and how the tendency for knee-jerk reactions has led to destruction of entire ecosystems of native plants, etc etc.... it was confirmed that my specimens are indeed of the invasive variety. Excerpts of the discussion and the specimen photos below, for your reference: "Your plant is Lonicera morrowii (Morrow's honeysuckle), a non-native species that was first introduced to MA for horticulture (it is from Asia). It can be very invasive. The native species of honeysuckle have very different flowers (examine web images of Lonicera canadensis and Lonicera villosa for comparison)." [Note: Those are two of the native species for New England] "The pink flowers are often the hybrid between Lonicera morrowii and Lonicera tatarica (called Lonicera X bella). "Lonicera morrowii and its hybrids do have hollow stems. [...] Our two native species of honeysuckles found in forested habitats have solid stems." TL;DR: Hollow stems = invasives!