Stabbing Pairwise Intersecting Disks by Four Points

• Published in: Discrete & computational geometry Vol. 70; no. 4; pp. 1751 - 1784
• Main Authors: Carmi, Paz, Katz, Matthew J., Morin, Pat
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-12-2023; Springer Nature B.V
• Subjects: Algorithms; Combinatorics; Computational Mathematics and Numerical Analysis; Computer science; Disks; Geometry; Mathematics; Mathematics and Statistics; Stabbings; Piercing set; Disks in the plane; Helly-type theorems; 68U05; 52C10
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-023-00567-0

In their seminal work, Danzer (Stud. Sci. Math. Hungar.
21
(1–2), 111–134 (1986)) and Stachó (Mat. Lapok
32
(1–3), 19–47 (1981–84)) established that every set
D
of pairwise intersecting disks in the plane can be stabbed by four points. Recently, Har-Peled et al. (Discrete Math.
344
(7), # 112403 (2021)) presented a relatively simple linear-time algorithm for finding five points that stab 
D
. We present an alternative (somewhat less involved) proof to the assertion that four points are sufficient to stab 
D
. Moreover, our proof is constructive and provides a simple linear-time algorithm for finding the stabbing points. As a warmup, we present a nearly-trivial linear-time algorithm with an elementary proof for finding five points that stab 
D
.