Piercing Diametral Disks Induced by Edges of Maximum Spanning Trees

Abstract

Let P be a set of points in the plane and let T be a maximum-weight spanning tree of P. For an edge (p, q), let $D_{pq}$ be the diametral disk induced by (p, q), i.e., the disk having the segment $\overline{pq}$ as its diameter. Let ${\mathcal{{D}}}_T$ be the set of the diametral disks induced by the edges of T. In this paper, we show that one point is sufficient to pierce all the disks in ${\mathcal{{D}}}_T$, thus, the set ${\mathcal{{D}}}_T$ is Helly. Actually, we show that the center of the smallest enclosing circle of P is contained in all the disks of ${\mathcal{{D}}}_T$, and thus the piercing point can be computed in linear time.

This work was partially supported by Grant 2016116 from the United States – Israel Binational Science Foundation.