Wow. I have a hoary PI vine growing up one particular tree. It's
maybe 3/4" to 1" thick. I've been chasing ground-level PI around with
roundup for a while, and was going to whack the main vine today.
Since it's raining, I didnt' get to it.

By what you say, it almost sounds like I'd be better to let the main
vine grow, so as not to force suckers all over the place...

Cut the vine just above a few stems of leaves. RoundUp those leaves and the
top of the vine. You don't want that puppy setting seed -- any suckers it
sends out can be mowed down or hit with RoundUp again later.

James