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The Density Formula: One Lemma to Bound Them All

Authors Michael Kaufmann , Boris Klemz , Kristin Knorr , Meghana M. Reddy , Felix Schröder , Torsten Ueckerdt

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ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Graphs and surfaces
Keywords
  • beyond-planar
  • density
  • fan-planar
  • fan-crossing
  • right-angle crossing
  • quasiplanar

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Abstract

We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing several applications: we prove tight upper bounds on the edge density of various beyond-planar graph classes, including so-called k-planar graphs with k = 1,2, fan-crossing/fan-planar graphs, k-bend RAC-graphs with k = 0,1,2, quasiplanar graphs, and k^+-real face graphs. In some cases (1-bend and 2-bend RAC-graphs and fan-crossing/fan-planar graphs), we thereby obtain the first tight upper bounds on the edge density of the respective graph classes. In other cases, we give new streamlined and significantly shorter proofs for bounds that were already known in the literature. Thanks to the Density Formula, all of our proofs are mostly elementary counting and mostly circumvent the typical intricate case analysis found in earlier proofs. Further, in some cases (simple and non-homotopic quasiplanar graphs), our alternative proofs using the Density Formula lead to the first tight lower bound examples.

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Michael Kaufmann, Boris Klemz, Kristin Knorr, Meghana M. Reddy, Felix Schröder, and Torsten Ueckerdt. The Density Formula: One Lemma to Bound Them All. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.7

Author Details

Michael Kaufmann
  • University of Tübingen, Germany
Boris Klemz
  • Universität Würzburg, Germany
Kristin Knorr
  • Freie Universität Berlin, Germany
Meghana M. Reddy
  • ETH Zürich, Switzerland
Felix Schröder
  • Technische Universität Berlin, Germany
  • Charles University, Prague, Czech Republic
Torsten Ueckerdt
  • Karlsruhe Institute of Technology, Germany

Funding

  • Schröder, Felix: supported by the Czech Science Foundation grant GAČR 23-04949X.
  • Ueckerdt, Torsten: supported by the Deutsche Forschungsgemeinschaft - 520723789.

References

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