Nodally 3-connected planar graphs and convex combination mappings

Abstract: A convex combination mapping of a planar graph is a plane mapping in which the external vertices are mapped to the corners of a convex polygon and every internal vertex is a proper weighted average of its neighbours. If a planar graph is nodally 3-connected or triangulated then every such mapping is an embedding (Tutte, Floater). We give a simple characterisation of nodally 3-connected planar graphs, and generalise the above result to any planar graph which admits any convex embedding.