What is this ivy?

What is this ivy and how do I eradicate from my garden?

 

Probably a Calystegia, definitely not a Hedera. Keep breaking it off and pulling at it, never let it remain in leaf and you should be able to starve it out.

 

Interesting, so does it have rhizomes like bamboo?

 

Slender, white inside and rubbery. Pull on one of the shoots or dig around beneath them and you will see.