I finally learned the name of the invasive vine that has been creeping into my yard from the neighbor's yard. It's Periwinkle. I thought Periwinkle was a small plant like violets! But a couple of posters on a garden site clued me in, and when I looked it up, there it was. I can't believe a week of searching "invasive vines" didn't help me find that out myself.
It is apparently hard to kill. Deep resilient spreading roots, leaves that don't absorb water, tolerant of many soils and sunlight. I even have some I put a trash barrel over last summer; the entire vine is white but won't die. They are surviving by nutrient spread from uncovered vines.
I weed-whacked them last week. That should weaken them for a while. But it seems there are only a few ways to eliminate them. Digging the roots out 2' down, covering them in black plastic for a couple years, and using herbicides.
Digging down that far needs heavy equipment and the space is too tight. I can't cover them all with plastic as they are among plants I want to keep. So that means herbicide. And it means an oil-based herbicide.
I try to stay organic. Not certified, but in general practice. So in the open areas it has invaded, I will spray carefully. I have a large cardboard box I have cut the bottom out of and made a small hole in the top. I will spray Triclopyr into the hole, let the box sit sit 5 minutes and move it to the next spot (wearing latex gloves and a mask).
Then I will cover the area with black plastic (I have large rolls of the stuff). For the individual plants in the flower beds, I will apply it with a 1" brush on leaves. And since I'm going that far, I will use the box on the poison ivy and the english ivy ("in for a penny, in for a pound").