Preassigning the shape of a face
David W. Barnette, Branko Grünbaum · Pacific Journal of Mathematics · 1970
THEOREM.Let G be a 3-connected planar graph, F a nonseparating w-circuit of G with nodes A lf -, A n (in a cyclic order of F), and let F' be a convex %-gon with vertices A'i, '", 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 A'i of P corresponds to the node Ai of G for i = 1,2, , w.