2022/10/21 by Aichholzer, Oswin, Hackl, Thomas, Löffler, Maarten +4
#Computational Geometry (cs.CG) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2210.12015
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 any set P of n points in general position 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.