Well, two questions really.

I'm working with the following graph transformation: take the directed graph

and make a new graph

where

and

.

My first question, does anyone recognize this tranformation? Does it have an existing name or anything I can look up?

Second, and more importantly, I need to know whether or not this transformation preserves planarity, but I don't know where to begin.

Thanks for any advice.

Edit: Well, I can show that it doesn't preserve planarity in all cases, but I still don't know what to call it.