THEOREM. Let G be a
3-connected planar graph, F a nonseparating n-circuit of G with nodes A1,⋯,An (in
a cyclic order of F), and let F′ be a convex n-gon with vertices A1′,⋯,An′ (in cyclic
order). Then there exists a 3-polytope P which realizes G, such that F′ is a
face of P and that the vertex Ai′ of P corresponds to the node Ai of G for
i = 1,2,⋯,n.