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Separable Drawings: Extendability and Crossing-Free Hamiltonian Cycles

Authors Oswin Aichholzer , Joachim Orthaber , Birgit Vogtenhuber

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LIPIcs.GD.2024.34.pdf
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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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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).

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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

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.

References

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