I found this just now. I hate to post a link to stackoverflow (cringe). The accepted answer might help you, or likely it probably won't. On looking at it, it reminds me of something you might see in a 3D modelling tool.

https://stackoverflow.com/questions/25512321/algorithm-for-a-planar-graph-game

Also this might have something:
http://cs.rkmvu.ac.in/~shreesh/media/planarity.pdf

You might be able to figure it out, looking at this C code, which I think you already found above. Lucky it is a BSD license.

https://github.com/graph-algorithms/edge-addition-planarity-suite

"Are you talking about the Numberphile video? Or the papers? If it's the video then as I said above it doesn't contain any "pseudo code". If you're talking about the papers then can I ask you a couple of questions about those algorithms?"

I was recommending you try to find a proper course on the subject, as they are much easier to understand than just text books. I can't help you understand this subject, because after reading a little about it, I couldn't see how I could put it to use, so my motivation to understand it went away. That github link looks pretty good though, if I was doing it, I would use that, as well as well as check out the other links posted.