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Strongly Hyperbolic Unit Disk Graphs

Authors Thomas Bläsius , Tobias Friedrich , Maximilian Katzmann , Daniel Stephan



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Author Details

Thomas Bläsius
  • Karlsruhe Institute of Technology, Germany
Tobias Friedrich
  • Hasso Plattner Institute, University of Potsdam, Germany
Maximilian Katzmann
  • Karlsruhe Institute of Technology, Germany
Daniel Stephan
  • GSV Algorithm Consulting UG (haftungsbeschränkt) & Co. KG, Potsdam, Germany

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Thomas Bläsius, Tobias Friedrich, Maximilian Katzmann, and Daniel Stephan. Strongly Hyperbolic Unit Disk Graphs. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 13:1-13:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.STACS.2023.13

Abstract

The class of Euclidean unit disk graphs is one of the most fundamental and well-studied graph classes with underlying geometry. In this paper, we identify this class as a special case in the broader class of hyperbolic unit disk graphs and introduce strongly hyperbolic unit disk graphs as a natural counterpart to the Euclidean variant. In contrast to the grid-like structures exhibited by Euclidean unit disk graphs, strongly hyperbolic networks feature hierarchical structures, which are also observed in complex real-world networks. We investigate basic properties of strongly hyperbolic unit disk graphs, including adjacencies and the formation of cliques, and utilize the derived insights to demonstrate that the class is useful for the development and analysis of graph algorithms. Specifically, we develop a simple greedy routing scheme and analyze its performance on strongly hyperbolic unit disk graphs in order to prove that routing can be performed more efficiently on such networks than in general.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graph algorithms
Keywords
  • hyperbolic geometry
  • unit disk graphs
  • greedy routing
  • hyperbolic random graphs
  • graph classes

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