Do NOT throw ivy in your compost bin. It will resurrect like a zombie plant and take over your garden overnight. Seriously, ivy is the Terminator of the plant world.
Bag it all up—every snapped piece, every stray leaf. Seal the bag tight and put it in the trash. Out of sight, out of root system.
How to get rid of ivy naturally on fences and walls
If you leave even a small piece on the ground, it might re-root and send a single, spiteful tendril back to the fence. Ivy holds grudges like a soap opera villain.