Drawing Graphs on the Torus

Let \(G\) be a \(2\)-connected graph with a toroidal rotation system given. An algorithm for constructing a straight line drawing with no crossings on a rectangular representation of the torus is presented. It is based on Read’s algorithm for constructing a planar layout of a \(2\)-connected graph with a planar rotation system. It is proved that the method always works. The complexity of the algorithm is linear in the number of vertices of \(G\).