A cubic planar graph gives a cell decomposition of a twosphere, whose dual complex is a triangulation. If I understand Plateau's laws here, a soap film gives a cell decomposition whose dual complex is a triangulation of a threesphere. Are there any combinatorial restrictions on the triangulations that can appear, as dual to a genuine soap film in $\mathbf{R}^3$?
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1$\begingroup$ Can you clarify, are you allowing the faces and regions of the soap bubbles to be nonsimply connected? For example there are (unstable) toroidal double bubbles satisfying Plateau's laws. One can form a dual complex abstractly, but it might not be a triangulation of S^3 (for example, there may be no vertices). torus.math.uiuc.edu/jms/Images/double $\endgroup$ – Ian Agol Jul 23 '16 at 15:19

$\begingroup$ Hi Ian. I'd be just as interested to find out there are combinatorial restrictions on foam with nonsimplyconnected bubbles. But the actual question that came up for me is even more restricted: if you draw a cubic planar graph on the boundary of a ball, what kind of minimal surfaces, with Plateau singularities and without interior regions, can you fill inside? $\endgroup$ – David Treumann Jul 23 '16 at 16:14