Testing Planarity of Geometric Automorphisms in Linear Time

It is a well-known result that testing a graph for planarity and, in the affirmative case, computing a planar embedding can be done in linear time. In this paper, we show that the same holds if additionally we require that the produced drawing be symmetric with respect to a given automorphism of the graph. This problem arises naturally in the area of automatic graph drawing, where symmetric and planar drawings are desired whenever possible

On Nearly Symmetric Drawings of Graphs

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