Flexibility of Polyhedral Embeddings of Graphs in Surfaces

Whitneyʹs 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→N0 such 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.