# Convex polyhedra, combinatorial types and Symmetry

Steinitz theorem says that combinatorial types of convex polyhedra is identified with 3-connected planar graph, called by a polyhedral graph.

A symmetry of polyhedral graph means that a vertex permutation( where the corresponding edges/faces permutations are also induced from vertices) preserving the combinatorial type. We can easily check that this is realized as a planar isotopy and reflection on 2-sphere. I want to say this is combinatorial symmetry and consider the group of combinatorial symmetries for a polyhedral graph.

Now, my question is the following.

Let us consider a polyhedral graph and the combinatorial symmetry group. Is there a Euclidean convex polyhedron of the polyhedral graph whose combinatorial symmetry group can be realized as Euclidean isometries?

For example, $K_4$ has the tetrahedral symmetry group. If one consider regular tetrahedra in Euclidean space, the all tetrahedral symmetries can be realized as Euclidean isometries.

Is this true for all polyhedral graph? .

Yes, this is true for all polyhedral graphs. See the following reference:

P. Mani, Automorphismen von polyedrischen Graphen, Mathematische Annalen Volume 192, Issue 4, pp 279-303, August 1971

My translation:

Theorem. For every graph $\mathfrak{Q}$ there is a three dimensional convex polyhedron $P$ with the properties:

(a) $\mathfrak{Q}$ is isomorphic to vertex-edge skeleton $\mathfrak{P}^1$ of $P$

(b) Every automorphism of $\mathfrak{P}^1$ is induced by a symmetry of the polyhedron $P$.

See this mathscinet link for an english summary of the result.