If English Ivy wasn't climbing a tree or wall, it won't be mature enough (in terms of type of growth produced) to have flowered and fruited (formed seeds), so you won't need to worry about berry contamination of landfill-bound yard waste. You could check with your municipal landfill about what they do or don't accept in the yard waste stream, but aside from local restrictions, generally, invasive plants should go into the trash. Since it's more sustainable to compost them, though, and since industrial-type composting facilities like the ones presumably used by municipal landfills will kill weed seeds and plant parts, it's probably low-risk to include them. Disposal would necessitate being more careful if the plants were invasive species new to our area, since an escape of these two widespread invasive plants wouldn't really have much of an impact on local ecosystems, as they are already inundated with these and other long-established invasive species.

If you're wary of the weed parts remaining viable enough to root once buried, just dry them in the sun (having removed the soil and/or cut them into smaller pieces first) for a week or two so they desiccate enough to not survive the addition to their compost pile.

Miri

You're welcome.