# Polyhedra Classification

The following is inspired by this question. From time to time I search the web for tables of polyhedra, but without much success. Part of the problem is that there are many non-equivalent questions that can be asked. For example:

• If $G$ is a planar graph where every edge is in a cycle, what extra conditions are needed so that $G$ realizable as a polyhedron?
• How many polyhedra with $e$ edges are there?
• If $G$ is a planar graph where every edge is in a cycle, what extra conditions are needed so that $G$ realizable as a polyhedron with regular faces?
• How many polyhedra with $e$ edges and regular faces are there?
• What are the obstructions to realizability?
• etc$\ldots$

Meta-question: Where are questions like these addressed?

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Here are three sources, specifically on convex polyhedra (and polytopes), followed (now) by one reference on nonconvex polyhedra. These by no means constitute a complete answer to your broad question.

The classic source is:

Branko Grünbaum. Convex Polytopes, 2nd ed. New York: Springer-Verlag, 2003. (Amazon link)

A modern treatment:

Günter M. Ziegler. Lectures on Polytopes. Graduate Texts in Mathematics 152. Springer-Verlag New York Berlin Heidelberg, Revised sixth printing 2006. (Springer link)

Finally, the integer sequence A000944 counts the number of convex polyhedra with $n$ vertices (and cites Grünbaum).

In response to Dima's question, let me add:

Branko Grünbaum. "Graphs of polyhedra; polyhedra as graphs." Discrete Mathematics. Volume 307, Issues 3–5, 6 February 2007, Pages 445–463. (Elsevier link)

He says, "The central obstacle to any coherent theory of polyhedra more general than the convex ones is the difficulty of defining precisely what objects should be awarded that designation."

Here is the abstract:

Relations between graph theory and polyhedra are presented in two contexts. In the first, the symbiotic dependence between 3-connected planar graphs and convex polyhedra is described in detail. In the second, a theory of nonconvex polyhedra is based on a graph-theoretic foundation. This approach eliminates the vagueness and inconsistency that pervade much of the literature dealing with polyhedra more general than the convex ones.

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I wonder if anyone looked at the non-convex case. –  Dima Pasechnik Mar 27 '13 at 14:14
Thanks! ...and nice pictures. –  Rodrigo A. Pérez Mar 29 '13 at 5:11

One more article (it is absent in the book of Grünbaum):

Bilinski, Stanko Über die Ordnungszahl der Klassen Eulerscher Polyeder. Arch. Math. 10, 180-186 (1959).

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