An algorithm for straight-line representation of simple planar graphs

Abstract

An algorithm is developed for drawing straight-line planar graphs which are isomorphic to a convex polyhedron and simple (i.e. a connected graph with no self-loops or multiple branches). The construction of such graphs is outlined in three stages. Stage 1 determines all the independent cycles of the graph plus one dependent cycle by Goldstein's algorithm. Stage 2 orders the cycles in such a way that they may be drawn in sequence with a given cycle as external. Stage 3 constructs the graph by successively assigning coordinates to the vertices which were placed in order according to the result of Stage 2. An iterative procedure is used for shifting the coordinates to eliminate crossings of edges. Results show that practically all these planar connected graphs up to 22 cycles can be drawn provided that the longest cycle is chosen as the external one. In addition, in the event of failure of the iteration procedure for graphs with a large number of cycles, a heuristic decision involving man-machine interaction is introduced to adjust the partially completed straight-line graph.