Intersection Graphs of Rays and Grounded Segments

  • Jean Cardinal
  • Stefan Felsner
  • Tillmann MiltzowEmail author
  • Casey Tompkins
  • Birgit Vogtenhuber
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10520)


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.



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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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Jean Cardinal
    • 1
  • Stefan Felsner
    • 2
  • Tillmann Miltzow
    • 3
    Email author
  • Casey Tompkins
    • 4
  • Birgit Vogtenhuber
    • 5
  1. 1.Université libre de Bruxelles (ULB)BrusselsBelgium
  2. 2.Institut für MathematikTechnische Universität Berlin (TU)BerlinGermany
  3. 3.Institute for Computer Science and ControlHungarian Academy of Sciences (MTA SZTAKI)BudapestHungary
  4. 4.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  5. 5.Institute of Software TechnologyGraz University of TechnologyGrazAustria

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