Flexibility of polyhedral embeddings of graphs in surfaces

Whitney's theorem states that 3-connected planar graphs admit essentially u nique embeddings in the plane. We generalize this result to embeddings of g raphs in arbitrary surfaces by showing that there is a function xi: N-0-->N -0 such that every 3-connected graph admits at most xi (g) combinatorially distinct embeddings of face-width greater than or equal to3 into surfaces w hose Elder genus is at most g. (C) 2001 Academic Press.