## Abstract

Given two distinct point sets P and Q in the plane, we say that Q blocks P if no two points of P are adjacent in any Delaunay triangulation of P ∪ Q. Aichholzer et al. (2013) showed that (the Delaunay triangulation of) any set P of n points in general position (that is, no three collinear and no four cocircular) can be blocked by 3

2 n points and that every set P of n points in convex position can be blocked by 5

4 n points. Moreover, they conjectured that, if P is in convex position, n blocking

points are sufficient and necessary. The necessity was recently shown by Biniaz (2021) who proved that every point set in general position requires n blocking points.

Here we investigate the variant, where blocking points can only lie outside of the convex hull of the given point set. We show that 5 4 n − O(1) such exterior-blocking points are sometimes necessary, even if the given point set is in convex position. As a consequence we obtain that, if the conjecture of Aichholzer et al. for the original setting was true, then minimal blocking sets of some point

configurations P would have to contain points inside of the convex hull of P .

2 n points and that every set P of n points in convex position can be blocked by 5

4 n points. Moreover, they conjectured that, if P is in convex position, n blocking

points are sufficient and necessary. The necessity was recently shown by Biniaz (2021) who proved that every point set in general position requires n blocking points.

Here we investigate the variant, where blocking points can only lie outside of the convex hull of the given point set. We show that 5 4 n − O(1) such exterior-blocking points are sometimes necessary, even if the given point set is in convex position. As a consequence we obtain that, if the conjecture of Aichholzer et al. for the original setting was true, then minimal blocking sets of some point

configurations P would have to contain points inside of the convex hull of P .

Original language | English |
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Title of host publication | Proc. 38th European Workshop on Computational Geometry (EuroCG 2022) |

Place of Publication | Perugia, Italy |

Pages | 9:1-9:7 |

Publication status | Published - 2022 |

Event | 38th European Workshop on Computational Geometry: EuroCG 2022 - Engineering Department of the University of Perugia, Perugia, Italy Duration: 14 Mar 2022 → 16 Mar 2022 https://eurocg2022.unipg.it/ |

### Conference

Conference | 38th European Workshop on Computational Geometry |
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Abbreviated title | EuroCG 2022 |

Country/Territory | Italy |

City | Perugia |

Period | 14/03/22 → 16/03/22 |

Internet address |

## Fields of Expertise

- Information, Communication & Computing