Symmetric embeddings of graphs

Let G be a graph, G′ an embedding of G as a straight 1-complex in Rⁿ, the real coordinate space of dimension n; let Φ be a group of transformations mapping Rⁿ to itself. If for every automorphism α of G we can find a member of Φ mapping G′ onto itself in such a way that it induces α in G′, we say that G′ is a Φ-symmetric embedding of G. In particular this paper discusses conditions for the existence of such an embedding when Φ is the group of autohomeomorphisms of Rⁿ or the group of invertible linear transformations in Rⁿ, and the graph is the complete graph K_m.