From planar graph layout to flat embeddings of 2-complexes

A question about 2-complexes, analogous to graph planarity, is whether they can be embedded in 3 dimensions. This is a difficult question. Also, whereas planar graphs always admit straightedge embeddings, it is known that 2-complexes may admit embeddings without admitting flatface embeddings. We consider three methods for straight-edge planar graph layout: barycentric maps (and convex-combination maps) which have already been studied in 3 dimensions, Read’s vertexdeletion method, and the De Fraysseix-Pach-Pollack method. We give examples showing that each method can fail. The last method is based on so-called monotone subdivisions, and not only are some subdivisions intrinsically non-monotone, but some monotone subdivisions will not lead to a flat embedding.