Which starts to be present when the potatoes start turning green.
Potatoes start turning green once they're exposed to light. That's because potatoes are actually just stems of the plant and are thus capable of growing a whole new plant.
So... If you prevent the potatoes from getting green but keeping them in the darkness, then your should be good. Not an advice of any kinds, just a botanist sharing love~
But, the term the GP is looking for is "stolon" I think.
When you cut up a potato and get 4 new plants it's like taking a cutting from a tree.
Keep in mind they also still grow "normally" from seeds; there are true "potato fruits" (completely inedible due to the solanine) produced from flowers that contain seeds which grow new potatoes. It's just not economical to grow them this way.
Here's a pretty cool thing to think about:
An apple is a grandchild of the tree.
Thus the saying an apple doesn't fall far from the tree would, correctly speaking, mean that the person is just like their grandparent and not just like their parent.