Soap bubble graphs

Last week I was in Prague as one of four invited speakers at the EuroGIGA Midterm Conference; EuroGIGA is a big multi-investigator European research project on graphs, geometry, and algorithms.

My talk there was entitled "Möbius transformations, power diagrams, Lombardi drawings, and soap bubbles", and it announced the results of two papers that I've recently uploaded to the arXiv: "Planar Lombardi drawings for subcubic graphs" (arXiv:1206.6142) and "The graphs of planar soap bubbles" (arXiv:1207.3761).

I think it's easier to explain what I'm doing in the soap bubble one, so let's start with that. The following picture could not possibly be a (two-dimensional) soap bubble, but why not?



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