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In this paper we show that if attention is restricted to polyhedral embeddings of graphs, no theorem analogous to the Duke interpolation theorem for 2-cell embeddings is true. We also give two interesting classes of graphs: (i) a class in which the members have polyhedral embeddings in the torus and also in orientable manifolds of arbitrarily high genus, (ii) and another in which the members have polyhedral embeddings in the projective plane and also in orientable and nonorientable manifolds of arbitrarily low Euler characteristic.
On the noninterpolation of polyhedral maps. (1994). Discrete Mathematics, 131(1), 211-219. https://doi.org/10.1016/0012-365X(94)90386-7