Dig out what you can. It doesn't resprout much from the roots, but will from the crown at the base of the stems, so get 6 inches or so of root with the removed tops. No need to dig out every root (aren't you relieved?)

Keep removing the top growth as fast as it resprouts and it will eventually exhaust its reserves of energy.

And it self-seeds easily in some areas, so patrol for them.

That is good news! Thank you for this! The other thing I failed to mention is that the one growing through the cement is actually coming from a plant that is in the yard next door. Does this one bear much hope, if next door do not want to remove it?

If you can cut the roots from that plant next door you can keep it from growing in your yard. That may mean using something like a spade to cut as deep as possible for several feet along the property line.

kimmq is kimmsr

The other thing I failed to mention is that the one growing through the
cement is actually coming from a plant that is in the yard next door.
Does this one bear much hope, if next door do not want to remove it?

That's when you use glyphosate? :)

Or dig along the edge of the cement and sever the roots, then clip the tops out. it is not a deep-rooted plant.

lol yes trying very hard to avoid that. It's in a tricky spot so I don't think digging under will be a possibility, although I may redo the whole area eventually. I think I'll try boiling water on cut roots before round up and see how that goes. Cutting the roots (very thick) is the challenge due to location - any recommended tools for cutting roots where there is little space for sawing?

There are specialty "crack weeders" ... sharp hooked or right angle blades.You pull up on the vines and slash as low as you can go in the crevices.

I'd use a cane knife (the sugar harvesting tool) , or a carpet knife. You need to have the blade sharpened on the inside of the hook.

Special crack weed tool:

Brilliant, thanks!