Graph Embedding with Minimum Depth and Maximum External Face

We present new linear time algorithms using the SPQR-tree data structure for computing planar embeddings of planar graphs optimizing certain distance measures. Experience with orthogonal drawings generated by the topology-shape-metrics approach shows that planar embeddings following these distance measures lead to improved quality of the final drawing in terms of bends, edge length, and drawing area.Given a planar graph, the algorithms compute the planar embedding with1the minimum depth among the set of all planar embeddings of G,2the external face of maximum size among the set of all planar embeddings of G,3the external face of maximum size among the set of all embeddings of G with minimum depth.