Abstract
For a planar graph G and a set \(\varPi \) of simple paths in G, we define a metro-map embedding to be a planar embedding of G and an ordering of the paths of \(\varPi \) along each edge of G. This definition of a metro-map embedding is motivated by visual representations of hypergraphs using the metro-map metaphor. In a metro-map embedding, two paths cross in a so-called vertex crossing if they pass through the vertex and alternate in the circular ordering around it.
We study the problem of constructing metro-map embeddings with the minimum number of crossing vertices, that is, vertices where paths cross. We show that the corresponding decision problem is NP-complete for general planar graphs but can be solved efficiently for trees or if the number of crossing vertices is constant. All our results hold both in a fixed and variable embedding settings.
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Notes
- 1.
To avoid parallel edges, paths of length two with the appropriate modifications would do equally well.
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Acknowledgments
We thank the organizers and the other participants of the 2017 Dagstuhl seminar “Scalable Set Visualization”, where this work started, in particular Robert Baker, Nan Cao, and Yifan Hu.
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Frank, F. et al. (2021). Using the Metro-Map Metaphor for Drawing Hypergraphs. In: Bureš, T., et al. SOFSEM 2021: Theory and Practice of Computer Science. SOFSEM 2021. Lecture Notes in Computer Science(), vol 12607. Springer, Cham. https://doi.org/10.1007/978-3-030-67731-2_26
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