The Yao Graph $Y_5$ is a Spanner

TL;DR

This paper proves that the Yao graph with five cones, $Y_5$, is a spanner with a specific stretch factor, completing the classification of Yao graphs as spanners for $k eq 3$, and shows that the Yao-Yao variant is not a spanner.

Contribution

It establishes that $Y_5$ is a spanner with a stretch factor of approximately 3.74, and that $Y_k$ is a spanner if and only if $k eq 3$, resolving an open problem.

Findings

$Y_5$ is a spanner with stretch factor ~3.74.

$Y_k$ is a spanner iff $k eq 3$.

$YY_5$ is not a spanner.

Abstract

In this paper we prove that $Y_5$, the Yao graph with five cones, is a spanner with stretch factor $\rho = 2+\sqrt{3} \approx 3.74$. Since $Y_5$ is the only Yao graph whose status of being a spanner or not was open, this completes the picture of the Yao graphs that are spanners: a Yao graph $Y_k$ is a spanner if and only if $k \geq 4$. We complement the above result with a lower bound of 2.87 on the stretch factor of $Y_5$. We also show that $YY_5$, the Yao-Yao graph with five cones, is not a spanner.