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Holes in Convex and Simple Drawings

Authors Helena Bergold , Joachim Orthaber , Manfred Scheucher , Felix Schröder

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Author Details

Helena Bergold
  • Institute of Computer Science, Freie Universität Berlin, Germany
  • TUM School of Computation, Information and Technology, Technische Universität München, Germany
Joachim Orthaber
  • Institute of Software Technology, Graz University of Technology, Austria
Manfred Scheucher
  • Institut für Mathematik, Technische Universität Berlin, Germany
Felix Schröder
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
  • Institute of Mathematics, Technische Universität Berlin, Germany

Funding

  • Bergold, Helena: Supported by DFG-Research Training Group "Facets of Complexity" (DFG-GRK 2434).
  • Orthaber, Joachim: Supported by the Austrian Science Fund (FWF) grant W1230.
  • Scheucher, Manfred: Supported by DFG Grant SCHE 2214/1-1.
  • Schröder, Felix: Supported by the GAČR Grant no. 23-04949X.

Cite AsGet BibTex

Helena Bergold, Joachim Orthaber, Manfred Scheucher, and Felix Schröder. Holes in Convex and Simple Drawings. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 5:1-5:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.GD.2024.5

Abstract

Gons and holes in point sets have been extensively studied in the literature. For simple drawings of the complete graph a generalization of the Erdős-Szekeres theorem is known and empty triangles have been investigated. We introduce a notion of k-holes for simple drawings and study their existence with respect to the convexity hierarchy. We present a family of simple drawings without 4-holes and prove a generalization of Gerken’s empty hexagon theorem for convex drawings. A crucial intermediate step will be the structural investigation of pseudolinear subdrawings in convex drawings.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Computational geometry
Keywords
  • simple topological graph
  • convexity hierarchy
  • k-gon
  • k-hole
  • empty k-cycle
  • Erdős-Szekeres theorem
  • Empty Hexagon theorem
  • planar point set
  • pseudolinear drawing

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References

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