Higher dimensional analog to planar graphs?

We usually say that a graph is planar if it can be embedded into 2-space s.t. no edges intersect. Here's a different way to describe the same situation: a graph is planar if it can be embedded into 2-space s.t. the edges form the boundaries of cells which nowhere overlap.

My question: Can you take this second definition and extend the idea to higher dimensions? For instance, allow the cells to be volumes in 3-space - like soap bubbles where the graph edges are the intersection of soap film walls.

Solution 1:

A planar graph is nothing other than a triangulation of the $2$-sphere (via stereographic projection), at least if one allows arbitrary polygons instead of just triangles, so a natural generalization to three dimensions is triangulations of the $3$-sphere where one allows arbitrary (convex) polyhedra instead of just simplices. One can of course talk about triangulations of arbitrary manifolds.

Solution 2:

Another generalization is along the genus axis, i.e., consider which graphs can be embedded into a 2d compact surface, such as an $n$-torus. This area is called topological graph theory.

Solution 3:

I'm not clear about your definition, for a graph below, what are cells and boundaries?

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