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Partitioning Complete Geometric Graphs on Dense Point Sets into Plane Subgraphs

Authors Adrian Dumitrescu , János Pach

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Author Details

Adrian Dumitrescu
  • Algoresearch L.L.C., Milwaukee, WI, USA
János Pach
  • Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • EPFL, Lausanne, Switzerland

Funding

Work on this paper was supported by ERC Advanced Grant no. 882971 "GeoScape." Work by the second named author was partially supported by NKFIH (National Research, Development and Innovation Office) grant K-131529.

Acknowledgements

We thank the anonymous reviewers for carefully reading the manuscript.

Cite AsGet BibTex

Adrian Dumitrescu and János Pach. Partitioning Complete Geometric Graphs on Dense Point Sets into Plane Subgraphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 9:1-9:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.GD.2024.9

Abstract

A complete geometric graph consists of a set P of n points in the plane, in general position, and all segments (edges) connecting them. It is a well known question of Bose, Hurtado, Rivera-Campo, and Wood, whether there exists a positive constant c < 1, such that every complete geometric graph on n points can be partitioned into at most cn plane graphs (that is, noncrossing subgraphs). We answer this question in the affirmative in the special case where the underlying point set P is dense, which means that the ratio between the maximum and the minimum distances in P is of the order of Θ(√n).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Convexity
  • complete geometric Graph
  • crossing Family
  • plane Subgraph

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