The Yao Graph $Y_5$ is a Spanner

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.