Flexibility of polyhedral embeddings of graphs in surfaces

Abstract

Whitney's theorem states that 3-connected planar graphs admit essentially unique embeddings in the plane. We generalize this result to embeddings of graphs in arbitrary surfaces by showing that there is a function ξ: ℕ0→ ℕ0such that every 3-connected graph admits at most ξ(g) combinatorially distinct embeddings of face-width ≥ 3 into surfaces whose Euler genus is at most g. © 2001 Academic Press.