The θ5-Graph is a Spanner

  • Prosenjit Bose
  • Pat Morin
  • André van Renssen
  • Sander Verdonschot
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8165)


Given a set of points in the plane, we show that the θ-graph with 5 cones is a geometric spanner with spanning ratio at most \(\sqrt{50 + 22\sqrt{5}}\) ≈ 9.960 . This is the first constant upper bound on the spanning ratio of this graph. The upper bound uses a constructive argument, giving a, possibly self-intersecting, path between any two vertices, whose length is at most \(\sqrt{50 + 22 \sqrt{5}}\) times the Euclidean distance between the vertices. We also give a lower bound on the spanning ratio of \(1\over2\) (11\(\sqrt{5}\)-17) ≈ 3.798.


Short Path Inductive Hypothesis Direct Edge Left Corner Total Path Length 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Prosenjit Bose
    • 1
  • Pat Morin
    • 1
  • André van Renssen
    • 1
  • Sander Verdonschot
    • 1
  1. 1.School of Computer ScienceCarleton UniversityCanada

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