, Volume 81, Issue 4, pp 1392–1415 | Cite as

On Plane Constrained Bounded-Degree Spanners

  • Prosenjit Bose
  • Rolf Fagerberg
  • André van RenssenEmail author
  • Sander Verdonschot


Let P be a finite set of points in the plane and S a set of non-crossing line segments with endpoints in P. The visibility graph of P with respect to S, denoted \(\mathord { Vis}(P,S)\), has vertex set P and an edge for each pair of vertices uv in P for which no line segment of S properly intersects uv. We show that the constrained half-\(\theta _6\)-graph (which is identical to the constrained Delaunay graph whose empty visible region is an equilateral triangle) is a plane 2-spanner of \(\mathord { Vis}(P,S)\). We then show how to construct a plane 6-spanner of \(\mathord { Vis}(P,S)\) with maximum degree \(6+c\), where c is the maximum number of segments of S incident to a vertex.


Plane spanners Bounded-degree Constraints Visibility graph 



This work is supported in part by the Natural Science and Engineering Research Council of Canada, Carleton University’s President’s 2010 Doctoral Fellowship, the Ontario Ministry of Research and Innovation, and the Danish Council for Independent Research, Natural Sciences, Grant DFF-1323-00247, and JST ERATO Grant No. JPMJER1201, Japan.


  1. 1.
    Bonichon, N., Gavoille, C., Hanusse, N., Ilcinkas, D.: Connections between theta-graphs, Delaunay triangulations, and orthogonal surfaces. In: Proceedings of the International Conference on Graph Theoretic Concepts in Computer Science, pp. 266–278 (2010)Google Scholar
  2. 2.
    Bonichon, N., Gavoille, C., Hanusse, N., Perkovic, L.: Plane spanners of maximum degree six. In: Proceedings of the International Colloquium on Automata, Languages and Programming, pp. 19–30 (2010)Google Scholar
  3. 3.
    Bonichon, N., Kanj, I., Perković, L., Xia, G.: There are plane spanners of degree 4 and moderate stretch factor. Discrete Comput Geom 53(3), 514–546 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bose, P., De Carufel, J.L., van Renssen, A.: Constrained generalized Delaunay graphs are plane spanners. In: Proceedings of the Computational Intelligence in Information Systems (CIIS 2016), Advances in Intelligent Systems and Computing, vol. 532, pp. 281–293 (2016)Google Scholar
  5. 5.
    Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: On plane constrained bounded-degree spanners. In: Proceedings of the 10th Latin American Symposium on Theoretical Informatics (LATIN 2012), Lecture Notes in Computer Science, vol. 7256, pp. 85–96 (2012)Google Scholar
  6. 6.
    Bose, P., Fagerberg, R., van Renssen, A., Verdonschot, S.: Optimal local routing on Delaunay triangulations defined by empty equilateral triangles. SIAM J. Comput. 44(6), 1626–1649 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bose, P., Keil, J.M.: On the stretch factor of the constrained Delaunay triangulation. In: Proceedings of the International Symposium on Voronoi Diagrams in Science and Engineering, pp. 25–31 (2006)Google Scholar
  8. 8.
    Bose, P., van Renssen, A.: Upper bounds on the spanning ratio of constrained theta-graphs. In: Proceedings of the 11th Latin American Symposium on Theoretical Informatics (LATIN 2014), Lecture Notes in Computer Science, vol. 8392, pp. 108–119 (2014)Google Scholar
  9. 9.
    Bose, P., Smid, M.: On plane geometric spanners: a survey and open problems. Comput. Geom. Theory Appl. CGTA 46(7), 818–830 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Clarkson, K.: Approximation algorithms for shortest path motion planning. In: Proceedings of the Annual ACM Symposium on Theory of Computing, pp. 56–65 (1987)Google Scholar
  11. 11.
    Das, G.: The visibility graph contains a bounded-degree spanner. In: Proceedings of the Canadian Conference on Computational Geometry, pp. 70–75 (1997)Google Scholar
  12. 12.
    Kanj, I., Perkovic, L., Turkoglu, D.: Degree four plane spanners: simpler and better. J. Comput. Geom. 8(2), 3–31 (2017)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
  2. 2.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdense MDenmark
  3. 3.School of Information TechnologiesUniversity of SydneySydneyAustralia

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