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Separable Drawings: Extendability and Crossing-Free Hamiltonian Cycles

Authors Oswin Aichholzer , Joachim Orthaber , Birgit Vogtenhuber

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Author Details

Oswin Aichholzer
  • Institute of Software Technology, Graz University of Technology, Austria
Joachim Orthaber
  • Institute of Software Technology, Graz University of Technology, Austria
Birgit Vogtenhuber
  • Institute of Software Technology, Graz University of Technology, Austria

Funding

  • Aichholzer, Oswin: Partially supported by the Austrian Science Fund (FWF) grant W1230.
  • Orthaber, Joachim: Supported by the Austrian Science Fund (FWF) grant W1230.
  • Vogtenhuber, Birgit: Partially supported by Austrian Science Fund within the collaborative DACH project Arrangements and Drawings as FWF project I 3340-N35.

Acknowledgements

We thank Alexandra Weinberger for fruitful discussions during the early stages of our research and we thank the anonymous referees for helpful comments on the paper.

Cite As Get BibTex

Oswin Aichholzer, Joachim Orthaber, and Birgit Vogtenhuber. Separable Drawings: Extendability and Crossing-Free Hamiltonian Cycles. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.34

Abstract

Generalizing pseudospherical drawings, we introduce a new class of simple drawings, which we call separable drawings. In a separable drawing, every edge can be closed to a simple curve that intersects each other edge at most once. Curves of different edges might interact arbitrarily. Most notably, we show that (1) every separable drawing of any graph on n vertices in the plane can be extended to a simple drawing of the complete graph K_n, (2) every separable drawing of K_n contains a crossing-free Hamiltonian cycle and is plane Hamiltonian connected, and (3) every generalized convex drawing and every 2-page book drawing is separable. Further, the class of separable drawings is a proper superclass of the union of generalized convex and 2-page book drawings. Hence, our results on plane Hamiltonicity extend recent work on generalized convex drawings by Bergold et al. (SoCG 2024).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
  • Theory of computation → Computational geometry
Keywords
  • Simple drawings
  • Pseudospherical drawings
  • Generalized convex drawings
  • Plane Hamiltonicity
  • Extendability of drawings
  • Recognition of drawing classes

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References

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