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Drawing Graphs with Circular Arcs and Right-Angle Crossings

Authors Steven Chaplick , Henry Förster , Myroslav Kryven, Alexander Wolff

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LIPIcs.SWAT.2020.21.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graphs and surfaces
  • Mathematics of computing → Combinatoric problems
Keywords
  • circular arcs
  • right-angle crossings
  • edge density
  • charging argument

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Abstract

In a RAC drawing of a graph, vertices are represented by points in the plane, adjacent vertices are connected by line segments, and crossings must form right angles. Graphs that admit such drawings are RAC graphs. RAC graphs are beyond-planar graphs and have been studied extensively. In particular, it is known that a RAC graph with n vertices has at most 4n-10 edges.
We introduce a superclass of RAC graphs, which we call arc-RAC graphs. A graph is arc-RAC if it admits a drawing where edges are represented by circular arcs and crossings form right angles. We provide a Turán-type result showing that an arc-RAC graph with n vertices has at most 14n-12 edges and that there are n-vertex arc-RAC graphs with 4.5n - O(√n) edges.

Cite As Get BibTex

Steven Chaplick, Henry Förster, Myroslav Kryven, and Alexander Wolff. Drawing Graphs with Circular Arcs and Right-Angle Crossings. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SWAT.2020.21

Author Details

Steven Chaplick
  • Maastricht University, the Netherlands
  • University of Würzburg, Germany
Henry Förster
  • University of Tübingen, Germany
Myroslav Kryven
  • University of Würzburg, Germany
Alexander Wolff
  • University of Würzburg, Germany

Funding

  • Kryven, Myroslav: M. Kryven acknowledges support from DFG grant WO 758/9-1.

Acknowledgements

We thank the reviewers of our paper for their very detailed comments, which helped us to improve the writing a lot.

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