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Saturated Simple and 2-simple Topological Graphs with Few Edges

  • Péter Hajnal
  • Alexander Igamberdiev
  • Günter Rote
  • André SchulzEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)

Abstract

A simple topological graph is a topological graph in which any two edges have at most one common point, which is either their common endpoint or a proper crossing. More generally, in a k-simple topological graph, every pair of edges has at most k common points of this kind. We construct saturated simple and 2-simple graphs with few edges. These are k-simple graphs in which no further edge can be added. We improve the previous upper bounds of Kynčl, Pach, Radoičić, and Tóth [Comput. Geom., 48, 2015] and show that there are saturated simple graphs on n vertices with only 7n edges and saturated 2-simple graphs on n vertices with 14.5n edges. As a consequence, 14.5n edges is also a new upper bound for k-simple graphs (considering all values of k). We also construct saturated simple and 2-simple graphs that have some vertices with low degree.

Keywords

Topological Graph Geometric Graph Large Face Black Edge Local Saturation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

The first author thanks Géza Tóth for presenting their inspiring results [5] in Szeged and for the encouragement during his investigation. This research was partially initiated at the EuroGIGA Workshop on Geometric Graphs in Münster, Germany, 2014, supported by the European Science Foundation (ESF).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Péter Hajnal
    • 1
  • Alexander Igamberdiev
    • 2
  • Günter Rote
    • 3
  • André Schulz
    • 4
    Email author
  1. 1.Bolyai InstituteUniversity of SzegedSzegedHungary
  2. 2.Institut für Mathematische Logik und GrundlagenforschungUniversität MünsterMünsterGermany
  3. 3.Institut für InformatikFreie Universität BerlinBerlinGermany
  4. 4.FernUniversität in HagenHagenGermany

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