Planar Graphs on Nonplanar Surfaces

It is shown that embeddings of planar graphs in arbitrary surfaces other than the 2-sphere have a special structure. It turns out that these embeddings can be described in terms of noncontractible curves in the surface, meeting the graph in at most two points (which may taken to be vertices of the graph). The close connection between the homology group of the surface and the planar graph embeddings is perhaps the most interesting aspect of this study. Some important consequences follow from these results. For example, any two embeddings of a planar graph in the same surface can be obtained from each other by means of simple local reembeddings very similar to Whitneyʹs switchings.