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:
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intersection graphs of grounded segments and intersection graphs of downward rays form the same graph class,
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not every intersection graph of rays is an intersection graph of downward rays, and
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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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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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