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On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs

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Abstract

Fan-planar graphs were recently introduced as a generalization of 1-planar graphs. A graph is fan-planar if it can be embedded in the plane, such that each edge that is crossed more than once, is crossed by a bundle of two or more edges incident to a common vertex. A graph is outer-fan-planar if it has a fan-planar embedding in which every vertex is on the outer face. If, in addition, the insertion of an edge destroys its outer-fan-planarity, then it is maximal outer-fan-planar. In this paper, we present a linear-time algorithm to test whether a given graph is maximal outer-fan-planar. The algorithm can also be employed to produce an outer-fan-planar embedding, if one exists. On the negative side, we show that testing fan-planarity of a graph is NP-complete, for the case where the rotation system (i.e., the cyclic order of the edges around each vertex) is given.

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Acknowledgments

This work started at the Bertinoro Workshop on Graph Drawing 2014. We thank the organizers and the participants of the workshop for the useful discussions on this topic. A preliminary version was presented at GD 2014 [5]. The work of Michael A. Bekos is implemented within the framework of the Action “Supporting Postdoctoral Researchers” of the Operational Program “Education and Lifelong Learning” (Action’s Beneficiary: General Secretariat for Research and Technology), and is co-financed by the European Social Fund (ESF) and the Greek State. Luca Grilli was partly supported by the MIUR Project AMANDA “Algorithmics for MAssive and Networked DAta”, prot. 2012C4E3KT_001. Seok-Hee Hong was partly supported by her ARC Future Fellowship and Humboldt Fellowship.

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Authors and Affiliations

  1. Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Tübingen, Germany

    Michael A. Bekos & Michael Kaufmann

  2. Department of Computer and Information Science, University of Konstanz, Konstanz, Germany

    Sabine Cornelsen

  3. Dipartimento di Ingegneria, Università degli Studi di Perugia, Perugia, Italy

    Luca Grilli

  4. School of Information Technologies, University of Sydney, Sydney, Australia

    Seok-Hee Hong

Authors
  1. Michael A. Bekos
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  2. Sabine Cornelsen
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  3. Luca Grilli
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  4. Seok-Hee Hong
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  5. Michael Kaufmann
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Correspondence to Michael A. Bekos.

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Bekos, M.A., Cornelsen, S., Grilli, L. et al. On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs. Algorithmica 79, 401–427 (2017). https://doi.org/10.1007/s00453-016-0200-5

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  • DOI: https://doi.org/10.1007/s00453-016-0200-5

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