  Let $C$ be the unit circle in $\mathbb{R}^2$. We can view $C$ as a plane
graph whose vertices are all the points on $C$, and the distance between any
two points on $C$ is the length of the smaller arc between them. We consider a
graph augmentation problem on $C$, where we want to place $k\geq 1$
\emph{shortcuts} on $C$ such that the diameter of the resulting graph is
minimized.
  We analyze for each $k$ with $1\leq k\leq 7$ what the optimal set of
shortcuts is. Interestingly, the minimum diameter one can obtain is not a
strictly decreasing function of~$k$. For example, with seven shortcuts one
cannot obtain a smaller diameter than with six shortcuts. Finally, we prove
that the optimal diameter is $2 + \Theta(1/k^{\frac{2}{3}})$ for any~$k$.
