The $\theta_5$-graph is a spanner

TL;DR

This paper proves that the $ heta_5$-graph is a geometric spanner with a bounded spanning ratio, providing the first constant upper bound and a lower bound on its efficiency.

Contribution

It establishes the first constant upper bound on the spanning ratio of the $ heta_5$-graph and provides bounds for its efficiency as a geometric spanner.

Findings

Upper bound on spanning ratio: approximately 9.960

Lower bound on spanning ratio: approximately 3.798

Constructive proof of path length between points

Abstract

Given a set of points in the plane, we show that the $\theta$-graph with 5 cones is a geometric spanner with spanning ratio at most $\sqrt{50 + 22 \sqrt{5}} \approx 9.960$. This is the first constant upper bound on the spanning ratio of this graph. The upper bound uses a constructive argument that gives a (possibly self-intersecting) path between any two vertices, of length 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 $\frac{1}{2}(11\sqrt{5} -17) \approx 3.798$.