Abstract. Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. This paper discusses symmetric drawings of oneconnected planar graphs. More specifically, we discuss planar (geometric) automorphisms, that is, automorphisms of a graph G that can be represented as symmetries of a planar drawing of G. Finding planar automorphisms is the first and most difficult step in constructing planar symmetric drawings of graphs. The problem of determining whether a given graph has a nontrivial geometric automorphism is NP-complete for general graphs. The two previous papers in this series have discussed the problem of drawing planar graphs with a maximum number of symmetries, for the restricted cases where the graph is triconnected and biconnected. This paper extends the previous results to cover planar graphs that are oneconnected. We present a linear time algorithm for drawing oneconnected planar graphs with a maximum number of symmetries. Key Words.

We propose a force-directed approach for drawing graphs in a nearly symmetric fashion. Our algorithm is built upon recent theoretical results on maximum symmetric subgraphs. Knowing the sequence of edge contractions sufficient for turning an asymmetric graph into a symmetric subgraph, our approach in symmetric drawing begins by drawing a graph's maximum symmetric subgraph using a force-directed method first; then the contracted edges are re-inserted back into the drawing. Considering symmetry as the underlying aesthetic criterion, our algorithm provides better drawings than the conventional spring algorithms, as our experimental results indicate.

This paper introduces a metric that measures symmetry in tree graphs, which allows for a statistical characterization of GP solutions by their architectural “shapes. ” A case study is given that applies this metric to 80.4 million trees to identify trends in GP runs. Results provide a first quantitative look at the dynamics of symmetry breaking.