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Associated with a graph $G$ and its automorphism group $\text{Aut}(G)$ (reflecting its combinatorial symmetries) are drawings in the plane with - eventually - one or more (geometric) symmetry groups.

For example the Petersen graph with automorphism group $S_5$ can be drawn with symmetry groups $D_1$, $D_5$ (dihedral groups), or cyclic group $\mathbb{Z}/5\mathbb{Z}$.

What can be said about possible combinations of geometric symmetry groups of graph drawings and their dependence on the automorphism group of the graph?

Is there for example always exactly one largest geometric symmetry group?

(Note that there are graphs with non-trivial automorphism group which don't admit a symmetric drawing in the plane at all:

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So on what else than the automorphism group does it depend whether there are geometric symmetries? Are there graphs with arbitrarily large automorphism group but without a symmetric drawing?)

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What do you assume about the "geometric symmetries" except that they move vertices and edges as expected? Are they to be plane isometries? Homeomorphisms? –  tomasz Jun 1 '13 at 22:57
    
I mean rotations and reflections. (What risk of confusion is there?) –  Hans Stricker Jun 1 '13 at 23:01
    
None anymore, I think. –  tomasz Jun 1 '13 at 23:17
    
I should have asked: What risk of confusion was there? –  Hans Stricker Jun 1 '13 at 23:21
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I am aware of Frucht's Theorem, but to be honest, I cannot see immediately how it answers my question. –  Hans Stricker Jun 1 '13 at 23:33

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