I have English Ivy and deal with it by ripping it out where ever I don't want it. If you mow over it, you may be seeing the foliage gone, but the vine can put down root systems along it. I have it in one border in the rock garden, I just keep it pulled and clipped away from perennials; but that's extra work that you may not want to hassel with. Doesn't seem to harm them, peonies been growing among that ivy for years and years.
Like I said, I just don't let it grow right up to the base of the plants. Some of my oldest hardy geraniums co-exist with the ivy as well, being surrounded by it; but having their own space. Wild columbine pushes itself up through every season. You can live with ivy if you keep a constant eye on it. This is my experience.

The longer it's there the worse it gets. It's root system is very dense and fibrous. It hogs all the water and nutrition in the soil and before long nothing else can live with it, though snakes adore it. If I were you, I would dig a trench along the fence, then spray everything on my side with Round Up, then dig what that doesn't kill. Then if you really want to keep it out, put some deep plastic edging or sheet metal into the trench several inches deep to form a root barrier. Of course, you'll still have to keep an eye on it so that it doesn't grow OVER your barrier. In my opinion, the only thing worse than english ivy is bamboo (and kudzu, but that's a southern plague).

It IS the bane of MY existence and has been ever since I STUPIDLY planted it 18 years ago. It's Hedera Helix. It grows UNDER the patio and comes out the other side, then goes under the fence where I have since planted many clematis vines, therefore cannot spray Round-Up or any other weed killer. I rip as much as possible when I can see it well - in Spring and Fall after everything else has died back - then I use a product called 'Vinex' (you can google it - I had to buy it online but I hear ACE hardware carries it). But you must be very careful with it because it will kill everything it touches including 100 year old trees (eventually). I use the little 'paintbrush' tip on the container and apply to the end of the vine - pulling it back as far as I can until I can no longer get any out of the ground - then paint Vinex on it (about a half-inch of the vine).

But if you are successful -- and it will not happen fast -- your neighbors will no longer have it either.

The other major mistake of my life was Boston Ivy. And Trumpet Vine. And Sweet Autumn Clematis - which grew under the house and came out the other side. I need to move.

Westy

I like the trench idea. Although, I'm sure my husband will NOT be happy to dig a 30 foot long trench. ;) The really stupid thing is we have a very ugly chicken wire fence and then there is about 2 feet and then my neighbors have their very ugly chicken wire fence and the ivy grows between the fences freely. I'm wondering if I should just take down our fence and then I can have better access to killing the ivy for good. Hmmmmm.....