Piercing Diametral Disks Induced by Edges of Maximum Spanning Tree

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 ${\mathcal{D}}_{\mathcal{T}}$ 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 ${\mathcal{D}}_{\mathcal{T}}$, thus, the set ${\mathcal{D}}_{\mathcal{T}}$ is Helly. Actually, we show that the center of the smallest enclosing circle of $P$ is contained in all the disks of ${\mathcal{D}}_{\mathcal{T}}$, and thus the piercing point can be computed in linear time.

7 pages, 4 figures