International Symposium on Graph Drawing and Network Visualization

Graph Drawing and Network Visualization pp 166-179

# Simultaneous Embeddings with Few Bends and Crossings

• Fabrizio Frati
• Michael Hoffmann
• Vincent Kusters
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

## Abstract

A simultaneous embedding with fixed edges (Sefe) of two planar graphs R and B is a pair of plane drawings of R and B that coincide when restricted to their common vertices and edges. We show that whenever R and B admit a Sefe, they also admit a Sefe in which every edge is a polygonal curve with few bends and every pair of edges has few crossings. Specifically: (1) if R and B are trees then one bend per edge and four crossings per edge pair suffice, (2) if R is a planar graph and B is a tree then six bends per edge and eight crossings per edge pair suffice, and (3) if R and B are planar graphs then six bends per edge and sixteen crossings per edge pair suffice. This improves on results by Grilli et al. (GD’14), who prove that nine bends per edge suffice, and by Chan et al. (GD’14), who prove that twenty-four crossings per edge pair suffice.

## Notes

### Acknowledgments

This research initiated at the Workshop on Geometry and Graphs, held at the Bellairs Research Institute in Barbados in March 2015. The authors thank the other participants for a stimulating atmosphere. Frati also wishes to thank Anna Lubiw and Marcus Schaefer for insightful ideas they shared during the research for [7].

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© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• Fabrizio Frati
• 1
• Michael Hoffmann
• 2
• Vincent Kusters
• 2
1. 1.Dipartimento di IngegneriaUniversity Roma TreRomeItaly
2. 2.Department of Computer ScienceETH ZürichZürichSwitzerland