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Intersection Graphs of Rays and Grounded Segments

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Book cover Graph-Theoretic Concepts in Computer Science (WG 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10520))

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Abstract

We consider several classes of intersection graphs of line segments in the plane and prove new equality and separation results between those classes. In particular, we show that:

  • intersection graphs of grounded segments and intersection graphs of downward rays form the same graph class,

  • not every intersection graph of rays is an intersection graph of downward rays, and

  • not every outer segment graph is an intersection graph of rays.

The first result answers an open problem posed by Cabello and Jejčič. The third result confirms a conjecture by Cabello. We thereby completely elucidate the remaining open questions on the containment relations between these classes of segment graphs. We further characterize the complexity of the recognition problems for the classes of outer segment, grounded segment, and ray intersection graphs. We prove that these recognition problems are complete for the existential theory of the reals. This holds even if a 1-string realization is given as additional input.

T. Miltzow—Supported by the ERC grant PARAMTIGHT: “Parameterized complexity and the search for tight complexity results”, no. 280152.

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Acknowledgments

This work was initiated during the Order & Geometry Workshop organized by Piotr Micek and the second author at the Gultowy Palace near Poznan, Poland, on September 14–17, 2016. We thank the organizers and attendees, who contributed to an excellent work atmosphere. Some of the problems tackled in this paper were brought to our attention during the workshop by Michal Lason. The first author also thanks Sergio Cabello for insightful discussions on these topics.

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Authors and Affiliations

  1. Université libre de Bruxelles (ULB), Brussels, Belgium

    Jean Cardinal

  2. Institut für Mathematik, Technische Universität Berlin (TU), Berlin, Germany

    Stefan Felsner

  3. Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary

    Tillmann Miltzow

  4. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    Casey Tompkins

  5. Institute of Software Technology, Graz University of Technology, Graz, Austria

    Birgit Vogtenhuber

Authors
  1. Jean Cardinal
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  2. Stefan Felsner
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  3. Tillmann Miltzow
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  4. Casey Tompkins
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  5. Birgit Vogtenhuber
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Corresponding author

Correspondence to Tillmann Miltzow .

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Editors and Affiliations

  1. Utrecht University, Utrecht, The Netherlands

    Hans L. Bodlaender

  2. Rheinisch-Westfälische Technische Hochschule Aachen, Aachen, Germany

    Gerhard J. Woeginger

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Cardinal, J., Felsner, S., Miltzow, T., Tompkins, C., Vogtenhuber, B. (2017). Intersection Graphs of Rays and Grounded Segments. In: Bodlaender, H., Woeginger, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2017. Lecture Notes in Computer Science(), vol 10520. Springer, Cham. https://doi.org/10.1007/978-3-319-68705-6_12

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  • DOI: https://doi.org/10.1007/978-3-319-68705-6_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-68704-9

  • Online ISBN: 978-3-319-68705-6

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