Stabbing Pairwise Intersecting Disks by Five Points

TL;DR

This paper presents a linear-time algorithm to find five points stabbing any set of pairwise intersecting disks and discusses bounds on the number of points needed to stab certain disk configurations.

Contribution

It introduces a deterministic linear-time algorithm for stabbing pairwise intersecting disks with five points and provides bounds on stabbing fewer disks with fewer points.

Findings

A linear-time algorithm finds five stabbing points for intersecting disks.

A set of 13 disks cannot be stabbed by three points.

Eight disks can be stabbed by at most three points.

Abstract

Suppose we are given a set $\mathcal{D}$ of $n$ pairwise intersecting disks in the plane. A planar point set $P$ stabs $\mathcal{D}$ if and only if each disk in $\mathcal{D}$ contains at least one point from $P$. We present a deterministic algorithm that takes $O(n)$ time to find five points that stab $\mathcal{D}$. Furthermore, we give a simple example of 13 pairwise intersecting disks that cannot be stabbed by three points. Moreover, we present a simple argument showing that eight disks can be stabbed by at most three points. This provides a simple-albeit slightly weaker-algorithmic version of a classical result by Danzer that such a set $\mathcal{D}$ can always be stabbed by four points.