Blocking Delaunay Triangulations from the Exterior

TL;DR

This paper studies the problem of blocking Delaunay triangulations with points outside the convex hull of a given set, showing that a certain number of exterior points are sometimes necessary, which impacts existing conjectures.

Contribution

It introduces the exterior-blocking variant and proves that more points outside the convex hull are sometimes needed, challenging previous assumptions about minimal blocking sets.

Findings

Exterior-blocking sometimes requires n/4 points

Blocking points outside convex hull can be necessary

Implications for existing conjectures on minimal blocking sets

Abstract

Given two distinct point sets $P$ and $Q$ in the plane, we say that $Q$ \emph{blocks} $P$ if no two points of $P$ are adjacent in any Delaunay triangulation of $P\cup Q$. Aichholzer et al. (2013) showed that any set $P$ of $n$ points in general position can be blocked by $\frac{3}{2}n$ points and that every set $P$ of $n$ points in convex position can be blocked by $\frac{5}{4}n$ 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 $\frac{5}{4}n-O(1)$ such \emph{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 $P$.