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Monotone Drawings of Graphs with Fixed Embedding

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Abstract

A drawing of a graph is a monotone drawing if for every pair of vertices u and v there is a path drawn from u to v that is monotone in some direction. In this paper we investigate planar monotone drawings in the fixed embedding setting, i.e., a planar embedding of the graph is given as part of the input that must be preserved by the drawing algorithm. In this setting we prove that every planar graph on n vertices admits a planar monotone drawing with at most two bends per edge and with at most 4n−10 bends in total; such a drawing can be computed in linear time and requires polynomial area. We also show that two bends per edge are sometimes necessary on a linear number of edges of the graph. Furthermore, we investigate subclasses of planar graphs that can be realized as embedding-preserving monotone drawings with straight-line edges. In fact, we prove that biconnected embedded planar graphs and outerplane graphs always admit such drawings, and describe linear-time drawing algorithms for these two graph classes.

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Notes

  1. In [2] points N μ ,E μ ,S μ ,W μ are denoted as p N (μ),p E (μ),p S (μ),p W (μ), respectively.

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Acknowledgements

We thank the anonymous reviewers for their valuable comments.

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Authors and Affiliations

  1. Università Roma Tre, Roma, Italy

    Patrizio Angelini & Vincenzo Roselli

  2. Università degli Studi di Perugia, Perugia, Italy

    Walter Didimo

  3. University of Arizona, Tucson, AZ, USA

    Stephen Kobourov

  4. Faculty of Informatics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

    Tamara Mchedlidze

  5. National Technical University of Athens, Athens, Greece

    Antonios Symvonis

  6. University of Lethbridge, Lethbridge, Canada

    Stephen Wismath

Authors
  1. Patrizio Angelini
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  2. Walter Didimo
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  3. Stephen Kobourov
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  4. Tamara Mchedlidze
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  5. Vincenzo Roselli
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  6. Antonios Symvonis
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  7. Stephen Wismath
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Corresponding author

Correspondence to Patrizio Angelini.

Additional information

A shorter version of this work appeared in the Proceedings of the 18th International Symposium on Graph Drawing (GD 2011). Research supported in part by the MIUR project AlgoDEEP prot. 2008TFBWL4 and by the ESF project 10-EuroGIGA-OP-003 GraDR “Graph Drawings and Representations”. Work on these results began at the 6th Bertinoro Workshop on Graph drawing. Discussion with other participants is gratefully acknowledged.

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Angelini, P., Didimo, W., Kobourov, S. et al. Monotone Drawings of Graphs with Fixed Embedding. Algorithmica 71, 233–257 (2015). https://doi.org/10.1007/s00453-013-9790-3

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  • DOI: https://doi.org/10.1007/s00453-013-9790-3

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