A flamethrower.

That's not "chemical."

Be well -- and good luck.

:)

If it's Japanese Honeysuckle, it can grow 15' up into the trees and send runners underground into your yard for 8 feet. I'd cut it back and pull up runners. I'm not "into" using weed killer, however in this case, I would carefully spray the each runner you cut with Roundup (not overspraying) and in time you'll eradicate the vine. Don't let it flower and go to seed. My husband and I have been fighting Japanese Honeysuckle for 3 1/2 years. Our neighbor thinks it's "pretty" and refuses to accept the invasive nature of the plant. Good luck.

I agree with the previous poster, "sowngrown". We've owned our home for a few short years, and initially thought the honeysuckle growing along the fence in the backyard was charming. HA!!! What a nightmare that stuff has turned out to be. It is our neighbor's honeysuckle...so we're in the same situation. The stuff has actually formed good-sized TRUNKS, which have grown through the chainlink fencing that separates our yard from our neighbor's yard. Repeated applications of RoundUp and removal of those horrible runners are the only things that have been effective. And the runners can be longer than 8 feet, btw. I started pulling one up in our backyard recently...and I kept pulling... and pulling. That runner actually ended in our neighbor's yard, and was at least 20 feet in length. I can't plant anything along our back fence at this point, because the honeysuckle has formed tightly packed root mats that I can't dig into. If I had to do it all over again, I'd pay for professional removal.
Good Luck.