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Simultaneous Embedding of Embedded Planar Graphs

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Algorithms and Computation (ISAAC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7074))

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Abstract

Given k planar graphs G 1,…,G k , deciding whether they admit a simultaneous embedding with fixed edges (Sefe ) and whether they admit a simultaneous geometric embedding (Sge ) are NP-hard problems, for k ≥ 3 and for k ≥ 2, respectively. In this paper we consider the complexity of Sefe and of Sge when the graphs G 1,…,G k have a fixed planar embedding. In sharp contrast with the NP-hardness of Sefe for three non-embedded graphs, we show that Sefe is polynomial-time solvable for three graphs with a fixed planar embedding. Furthermore, we show that, given k embedded planar graphs G 1,…,G k , deciding whether a Sefe of G 1,…,G k exists and deciding whether an Sge of G 1,…,G k exists are NP-hard problems, for k ≥ 14 and k ≥ 13, respectively.

Work partially supported by the MIUR, project AlgoDEEP 2008TFBWL4, and by the ESF project 10-EuroGIGA-OP-003 “Graph Drawings and Representations”.

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Author information

Authors and Affiliations

  1. Dipartimento di Informatica e Automazione, Università Roma Tre, Italy

    Patrizio Angelini, Giuseppe Di Battista & Fabrizio Frati

  2. School of Information Technologies, The University of Sydney, Australia

    Fabrizio Frati

Authors
  1. Patrizio Angelini
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  2. Giuseppe Di Battista
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  3. Fabrizio Frati
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Editor information

Editors and Affiliations

  1. Chuo University, Kasuga, Bunkyo-ku, 112-8551, Tokyo, Japan

    Takao Asano

  2. Gunma University, 1-5-1 Tenjin-Cho, 376-8515, Kiryu-Shi, Japan

    Shin-ichi Nakano

  3. Japan Advanced Institute of Science and Technology, 1-1 Asahidai, 923-1292, Nomi, Ishikawa, Japan

    Yoshio Okamoto

  4. Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, 152-8552, Tokyo, Japan

    Osamu Watanabe

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Angelini, P., Di Battista, G., Frati, F. (2011). Simultaneous Embedding of Embedded Planar Graphs. In: Asano, T., Nakano, Si., Okamoto, Y., Watanabe, O. (eds) Algorithms and Computation. ISAAC 2011. Lecture Notes in Computer Science, vol 7074. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25591-5_29

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  • DOI: https://doi.org/10.1007/978-3-642-25591-5_29

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-25590-8

  • Online ISBN: 978-3-642-25591-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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