An Algorithm for Drawing a Graph Symmetrically

A technique is described that produces symmetric drawings of a graph G in the plane. The method requries that G have a non-trivial automorphism group Aut(G), and that a non-trivial symmetry g ∈ Aut(G) has been selected in some way. The technique is found to be very effective in practice. It forms a part of the Groups & Graphs** software package. 1. Symmetric Cycles Let G be a connected undirected simple graph on n vertices V (G). If u, v ∈ V (G), then u→ v means that u is adjacent to v (and so also v → u). The reader is referred to [1] for other graph-theoretic terminology. Let Aut(G) denote the automorphism group of G. Given a permutation g ∈ Aut(G), it is often possible to draw G is such a way that the symmetry g is evident in the drawing. For example, there are 3 well known drawings of the Petersen graph, each illustrating a different symmetry. Fig. 1, 3 views of the Petersen graph * This work was supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada. ** Groups & Graphs is available on the internet, from http://bkocay.cs.umanitoba.ca/G&G