Abstract
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.
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Bose, P., Morin, P., van Renssen, A., Verdonschot, S. (2013). The θ 5-Graph is a Spanner. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds) Graph-Theoretic Concepts in Computer Science. WG 2013. Lecture Notes in Computer Science, vol 8165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45043-3_10
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DOI: https://doi.org/10.1007/978-3-642-45043-3_10
Publisher Name: Springer, Berlin, Heidelberg
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