Document Open Access Logo

The Spanning Ratio of the Directed Θ₆-Graph Is 5

Authors Prosenjit Bose , Jean-Lou De Carufel , John Stuart , Darryl Hill

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2026.20.pdf
  • Filesize: 1.2 MB
  • 18 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
  • Theory of computation → Computational geometry
  • Theory of computation → Shortest paths
Keywords
  • Geometric Spanners
  • Theta Graphs
  • Directed Theta Graphs
  • Spanning Ratio
  • Computational Geometry

Metrics

Abstract

Given a finite set P ⊂ ℝ², the directed Theta-6 graph, denoted Θ₆(P), is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The Θ₆(P)-graph is defined as follows: the plane around each point u ∈ P is partitioned into 6 equiangular cones with apex u, and in each cone, u is joined to the point whose projection on the bisector of the cone is closest. Equivalently, the Θ₆(P)-graph contains an edge from u to v exactly when the interior of ∇_u^v is disjoint from P, where ∇_u^v is the unique equilateral triangle containing u on a corner, v on the opposite side, and whose sides are parallel to the cone boundaries. It was previously shown that the spanning ratio of the Θ₆(P)-graph is between 4 and 7 in the worst case (Akitaya, Biniaz, and Bose Comput. Geom., 105-106:101881, 2022). We close this gap by showing a tight spanning ratio of 5. This is the first tight bound proven for the spanning ratio of any Θ_k(P)-graph. Our lower bound models a long path by mapping it to a converging series. Our upper bound proof uses techniques novel to the area of spanners. We use linear programming to prove that among several candidate paths, there exists a path satisfying our bound.

Cite As Get BibTex

Prosenjit Bose, Jean-Lou De Carufel, John Stuart, and Darryl Hill. The Spanning Ratio of the Directed Θ₆-Graph Is 5. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.20

Author Details

Prosenjit Bose
  • School of Computer Science, Carleton University, Ottawa, Canada
Jean-Lou De Carufel
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
John Stuart
  • School of Electrical Engineering and Computer Science, University of Ottawa, Canada
Darryl Hill
  • School of Computer Science, Carleton University, Ottawa, Canada

References

  1. Hugo A. Akitaya, Ahmad Biniaz, and Prosenjit Bose. On the spanning and routing ratios of the directed Θ_6-graph. Comput. Geom., 105-106:101881, 2022. URL: https://doi.org/10.1016/j.comgeo.2022.101881.
  2. Luis Barba, Prosenjit Bose, Jean-Lou De Carufel, André van Renssen, and Sander Verdonschot. On the stretch factor of the theta-4 graph. In WADS, volume 8037 of Lecture Notes in Computer Science, pages 109-120. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-40104-6_10.
  3. Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse, and Ljubomir Perkovic. Tight stretch factors for L_1- and L_∞-Delaunay triangulations. Comput. Geom., 48(3):237-250, 2015. URL: https://doi.org/10.1016/j.comgeo.2014.10.005.
  4. Prosenjit Bose, Jean-Lou De Carufel, Darryl Hill, and Michiel Smid. On the spanning and routing ratio of the directed theta-four graph. Discret. Comput. Geom., 71(3):872-892, 2024. URL: https://doi.org/10.1007/s00454-023-00597-8.
  5. Prosenjit Bose, Jean-Lou De Carufel, Darryl Hill, and John Stuart. The spanning ratio of the directed θ₆-graph is 5, 2026. URL: https://arxiv.org/abs/2603.09048.
  6. Prosenjit Bose, Jean-Lou De Carufel, Pat Morin, André van Renssen, and Sander Verdonschot. Towards tight bounds on theta-graphs: More is not always better. Theor. Comput. Sci., 616:70-93, 2016. URL: https://doi.org/10.1016/j.tcs.2015.12.017.
  7. Prosenjit Bose, Jean-Lou De Carufel, and Sandrine Njoo. The exact spanning ratio of the parallelogram Delaunay graph. Theor. Comput. Sci., 1037:115152, 2025. URL: https://doi.org/10.1016/j.tcs.2025.115152.
  8. Prosenjit Bose and André van Renssen. Upper bounds on the spanning ratio of constrained theta-graphs. In LATIN, volume 8392 of Lecture Notes in Computer Science, pages 108-119. Springer, 2014. URL: https://doi.org/10.1007/978-3-642-54423-1_10.
  9. Paul Chew. There is a planar graph almost as good as the complete graph. In SCG, pages 169-177. ACM, 1986. URL: https://doi.org/10.1145/10515.10534.
  10. Paul Chew. There are planar graphs almost as good as the complete graph. J. Comput. Syst. Sci., 39(2):205-219, 1989. URL: https://doi.org/10.1016/0022-0000(89)90044-5.
  11. K. Clarkson. Approximation algorithms for shortest path motion planning. In Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, STOC '87, pages 56-65, New York, NY, USA, 1987. Association for Computing Machinery. URL: https://doi.org/10.1145/28395.28402.
  12. David P. Dobkin, Steven J. Friedman, and Kenneth J. Supowit. Delaunay graphs are almost as good as complete graphs. Discret. Comput. Geom., 5:399-407, 1990. URL: https://doi.org/10.1007/BF02187801.
  13. J. Mark Keil and Carl A. Gutwin. Classes of graphs which approximate the complete Euclidean graph. Discret. Comput. Geom., 7(1):13-28, December 1992. URL: https://doi.org/10.1007/BF02187821.
  14. Anna Lubiw and Debajyoti Mondal. Construction and local routing for angle-monotone graphs. J. Graph Algorithms Appl., 23(2):345-369, 2019. URL: https://doi.org/10.7155/jgaa.00494.
  15. Giri Narasimhan and Michiel Smid. Geometric Spanner Networks. Cambridge University Press, 2007. URL: https://doi.org/10.1017/CBO9780511546884.
  16. Ljubomir Perkovic, Michael Dennis, and Duru Türkoglu. The stretch factor of hexagon-Delaunay triangulations. J. Comput. Geom., 12(2):86-125, 2021. URL: https://doi.org/10.20382/jocg.v12i2a5.
  17. J. Ruppert and R. Seidel. Approximating the D-dimensional complete Euclidean graph. In Proceedings of the 3rd Annual Canadian Conference on Computational Geometry, pages 207-210, 1991. URL: https://cccg.ca/proceedings/1991/paper50.pdf.
  18. John Stuart, Jean-Lou De Carufel, and Prosenjit Bose. The exact routing and spanning ratio of arbitrary triangle Delaunay graphs. In Rahnuma Islam Nishat, editor, Proceedings of the 36th Canadian Conference on Computational Geometry, CCCG 2024, Brock University, St. Catharines, Canada, July 17-19, 2024, pages 83-89, 2024. URL: https://doi.org/10.48550/arXiv.2506.12625.
  19. André van Renssen, Yuan Sha, Yucheng Sun, and Sampson Wong. The tight spanning ratio of the rectangle Delaunay triangulation. In ESA, volume 274 of LIPIcs, pages 99:1-99:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ESA.2023.99.
  20. Ge Xia. The stretch factor of the Delaunay triangulation is less than 1.998. SIAM J. Comput., 42(4):1620-1659, 2013. URL: https://doi.org/10.1137/110832458.
  21. Ge Xia and Liang Zhang. Toward the tight bound of the stretch factor of Delaunay triangulations. In Proceedings of the 23rd Annual Canadian Conference on Computational Geometry, Toronto, Ontario, Canada, August 10-12, 2011, 2011. URL: http://www.cccg.ca/proceedings/2011/papers/paper57.pdf.
  22. Andrew Chi-Chih Yao. On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM J. Comput., 11(4):721-736, 1982. URL: https://doi.org/10.1137/0211059.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact