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Drawing Shortest Paths in Geodetic Graphs

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Graph Drawing and Network Visualization (GD 2020)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 12590))

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Abstract

Motivated by the fact that in a space where shortest paths are unique, no two shortest paths meet twice, we study a question posed by Greg Bodwin: Given a geodetic graph G, i.e., an unweighted graph in which the shortest path between any pair of vertices is unique, is there a philogeodetic drawing of G, i.e., a drawing of G in which the curves of any two shortest paths meet at most once? We answer this question in the negative by showing the existence of geodetic graphs that require some pair of shortest paths to cross at least four times. The bound on the number of crossings is tight for the class of graphs we construct. Furthermore, we exhibit geodetic graphs of diameter two that do not admit a philogeodetic drawing.

This research began at the Graph and Network Visualization Workshop 2019 (GNV’19) in Heiligkreuztal. S. C. is funded by the German Research Foundation DFG – Project-ID 50974019 – TRR 161 (B06). M. H. is supported by the Swiss National Science Foundation within the collaborative DACH project Arrangements and Drawings as SNSF Project 200021E-171681. S. K. is supported by NSF grants CCF-1740858, CCF-1712119, and DMS-1839274.

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Authors and Affiliations

  1. University of Konstanz, Konstanz, Germany

    Sabine Cornelsen

  2. University of Tübingen, Tübingen, Germany

    Maximilian Pfister, Henry Förster & Thomas Schneck

  3. University of Osnabrück, Osnabrück, Germany

    Martin Gronemann

  4. Department of Computer Science, ETH Zürich, Zürich, Switzerland

    Michael Hoffmann

  5. Department of Computer Science, University of Arizona, Tucson, USA

    Stephen Kobourov

Authors
  1. Sabine Cornelsen
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  2. Maximilian Pfister
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  3. Henry Förster
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  4. Martin Gronemann
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  5. Michael Hoffmann
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  6. Stephen Kobourov
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  7. Thomas Schneck
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Correspondence to Sabine Cornelsen .

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Editors and Affiliations

  1. LaBRI, University of Bordeaux, Talence, France

    David Auber

  2. Charles University, Prague, Czech Republic

    Pavel Valtr

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Cornelsen, S. et al. (2020). Drawing Shortest Paths in Geodetic Graphs. In: Auber, D., Valtr, P. (eds) Graph Drawing and Network Visualization. GD 2020. Lecture Notes in Computer Science(), vol 12590. Springer, Cham. https://doi.org/10.1007/978-3-030-68766-3_26

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  • DOI: https://doi.org/10.1007/978-3-030-68766-3_26

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-68765-6

  • Online ISBN: 978-3-030-68766-3

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