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Variants of the Segment Number of a Graph

  • Yoshio Okamoto
  • Alexander Ravsky
  • Alexander WolffEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11904)

Abstract

The segment number of a planar graph is the smallest number of line segments whose union represents a crossing-free straight-line drawing of the given graph in the plane. The segment number is a measure for the visual complexity of a drawing; it has been studied extensively.

In this paper, we study three variants of the segment number: for planar graphs, we consider crossing-free polyline drawings in 2D; for arbitrary graphs, we consider crossing-free straight-line drawings in 3D and straight-line drawings with crossings in 2D. We first construct an infinite family of planar graphs where the classical segment number is asymptotically twice as large as each of the new variants of the segment number. Then we establish the \(\exists \mathbb {R}\)-completeness (which implies the NP-hardness) of all variants. Finally, for cubic graphs, we prove lower and upper bounds on the new variants of the segment number, depending on the connectivity of the given graph.

Notes

Acknowledgments

We thank the organizers and participants of the 2019 Dagstuhl seminar “Beyond-planar graphs: Combinatorics, Models and Algorithms”. In particular, we thank Günter Rote and Martin Gronemann for suggestions that led to some of this research. We also thank Carlos Alegría. We thank our reviewers for an idea that improved the bound in Proposition 5, for suggesting the statement of Lemma 1, and for many other helpful comments.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of Electro-CommunicationsChōfuJapan
  2. 2.RIKEN Center for Advanced Intelligence ProjectTokyoJapan
  3. 3.Pidstryhach Institute for Applied Problems of Mechanics and MathematicsNational Academy of Sciences of UkraineLvivUkraine
  4. 4.Universität WürzburgWürzburgGermany

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