Hi Jamie,
Yes giffgaff is ultimately owned by telefonica, as we make clear in the 'about us' section of this site. So the tree you've got below is accurate although obviously not fully complete, as telefonica owns dozens if not hundreds of companies around the world.
giffgaff is legally and operationally a separate company, with our own record at Companies House, it's just that the funding to create giffgaff came from O2 telefonica, and we run our service on the O2 network.
