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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.

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  • $\begingroup$ Intuitively it seems that such "cell" construction would have bounded doubling dimensions, whereas planar graphs can have unbounded doubling dimension $\endgroup$ – Yaroslav Bulatov Feb 18 '11 at 21:29
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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.

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  • $\begingroup$ When you say "triangulation", do you mean representing vertices of a graph as points in the space? I have trouble imagining how tree structured-graphs fit into this construction $\endgroup$ – Yaroslav Bulatov Feb 18 '11 at 21:32
  • $\begingroup$ @Yaroslav: yes. See my comment to your answer. If this isn't a good definition for you, pretend I'm actually talking about CW-complexes. $\endgroup$ – Qiaochu Yuan Feb 18 '11 at 21:45
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    $\begingroup$ I think Qiaochu means tesselating. $\endgroup$ – John Berryman Feb 18 '11 at 21:46
  • $\begingroup$ ... although that might still not be precise here... $\endgroup$ – John Berryman Feb 18 '11 at 21:47
  • $\begingroup$ Tangent: One of the proofs of Sylvester-Gallai theorem uses this planar graph representation. $\endgroup$ – Aryabhata Feb 18 '11 at 22:36
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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.

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  • $\begingroup$ That's kind a interesting. $\endgroup$ – John Berryman Feb 20 '11 at 4:04
  • $\begingroup$ @John: Perhaps a better analog of planar graphs would use geodesic embededdings, not merely topological ones. That is, the edges would be geodesics. I don't know whether topological graph theory handles this nor whether there is a "Riemannian graph theory" out there. I did find the paper "A Riemannian approach to graph embedding", dx.doi.org/10.1016/j.patcog.2006.05.031 , which seems to contains some references for this approach. $\endgroup$ – lhf Feb 22 '11 at 11:09
  • $\begingroup$ perhaps, but when we start talking about geodesics, we're getting away from the more pure topological questions I was interested in. $\endgroup$ – John Berryman Feb 22 '11 at 14:21
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I'm not clear about your definition, for a graph below, what are cells and boundaries?


(source: yaroslavvb.com)

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    $\begingroup$ There is one infinite cell, and its boundary consists of every edge. $\endgroup$ – Qiaochu Yuan Feb 18 '11 at 21:20
  • $\begingroup$ what Qiaochu said $\endgroup$ – John Berryman Feb 18 '11 at 21:22
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Another -- I think more natural -- concept is the (PL)-embeddability of $k$-dimensional simplicial complexes into $R^d$, for certain pairs $k,d$. The case $k=1$, $d=2$ is graph planarity.

There are many recent algorithmic result on this, including polynmiality in the metastable dimension range ($d\geq 3(k+1)/2$) with fixed dimensions, NP-hardness (and probably undecidability) for larger $k$, and proven undecidability for $k=d$.

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