Misunderstanding Embeddings, and Whitney Flips

So I should correct something that I posted a while ago about embedding 3-connected graphs. The most obvious examples of this class of graph are polyhedra - tetrahedra, cubes, etc - and maybe it's obvious that there is only one way to embed these in the plane. So in that sense, 'enumerating' the embeddings for these graphs is quite easy ... there's just one to count.

Of course, this embedding can be drawn with any of its cycles as an outer face; this is what gave rise to the different looking drawings. I guess the way to think about it is that the embedding is on the surface of a sphere - where there is no 'privileged' face to call the outer face - and that the drawing on the plane just picks one of the faces to 'squash flat' as I put it back in (wow) 2011.



Anyway - on to Whitney flips! There are a class of graphs that can be embedded in the plane in different ways (that is, the combinatorial map is different for the same outer face). These are subject to a Whitney flip, named after the mathematician who laid the foundation for matroids among other things. As an example see the image above.

The only graphs that have this flip are ones with a 2-vertex cutset - the top and bottom vertices in the image. Of course, finding these is a whole other problem, which I may or may not get around to describing...


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