Shortcuts for the circle
Let C be the unit circle in R2. 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⩾1 shortcuts on C such that the diameter of the resulting graph is minimized. We analyze for each k with 1⩽k⩽7 what the optimal set of