## Abstract

Let C be the unit circle in 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⩾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 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+Θ([formula presented]) for any k.

Original language | English |
---|---|

Pages (from-to) | 37-54 |

Number of pages | 18 |

Journal | Computational Geometry |

Volume | 79 |

DOIs | |

Publication status | Published - 1 Feb 2019 |

## Keywords

- Geometric network
- Graph augmentation
- Graph diameter