My neighbour put in some ivy a few years ago, and it is migrating into
my yard. It actually looks nice in a small wooded area I have, and
tumbling over a stone wall separating the wooded area from the lawn, but
now it is starting into the lawn.

We tried pulling it up, but it has fairly strong roots, and that is
obviously only a temporary fix.

I have some Kleenup vegetation killer in my arsenal, but the label on
that says applying it to the leaves will kill everything, including the
roots, and I'm thinking it might kill everything back into my
neighbour's yard, while I would be happy just to control the ivy so it
stays out of my grass.

Is there an easy effective way to control this before it gets across my
lawn into my garden? I have some other invasive plantings, like bee
balm and mint, that I have been able to control with barriers, but this
ivy just goes up and over the barrier.