Abstract
Suppose we are given a set D of n pairwise intersecting disks in the plane. A planar point set P stabs D if and only if each disk in D contains at least one point from P. We present a deterministic algorithm that takes O(n) time to find five points that stab 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 D can always be stabbed by four points.
| Original language | English |
|---|---|
| Article number | 112403 |
| Journal | Discrete Mathematics |
| Volume | 344 |
| Issue number | 7 |
| DOIs | |
| State | Published - Jul 2021 |
Bibliographical note
Publisher Copyright:© 2021 Elsevier B.V.
Funding
A preliminary version appeared as S.Har-Peled, H.Kaplan, W.Mulzer, L.Roditty, P.Seiferth, M.Sharir, and M.Willert. Stabbing Pairwise Intersecting Disks by Five Points. Proc.29th ISAAC, pp.50:1–50:12. SHP was supported by NSF, USA AF awards CCF-1421231, and CCF-1217462. WM was supported by DFG, Germany grant MU/3501/1 and ERCSTG 757609. PS was supported by DFG, Germany grant MU/3501/1. MS was supported by ISF, Israel grant 892/13 and 260/18, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Blavatnik Research Fund in Computer Science at Tel Aviv University, Israel. HK was supported by ISF, Israel grants 1595-19 and the Blavatnik Family Foundation, USA. Work on this paper was supported in part by grants 1367/2016 and 1161/2011 from the German–Israeli Science Foundation (GIF), Israel. A preliminary version appeared as S.Har-Peled, H.Kaplan, W.Mulzer, L.Roditty, P.Seiferth, M.Sharir, and M.Willert. Stabbing Pairwise Intersecting Disks by Five Points. Proc.29th ISAAC, pp.50:1–50:12. SHP was supported by NSF, USA AF awards CCF-1421231 , and CCF-1217462 . WM was supported by DFG, Germany grant MU/3501/1 and ERC STG 757609 . PS was supported by DFG, Germany grant MU/3501/1 . MS was supported by ISF, Israel grant 892/13 and 260/18 , by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Blavatnik Research Fund in Computer Science at Tel Aviv University, Israel . HK was supported by ISF, Israel grants 1595-19 and the Blavatnik Family Foundation, USA . Work on this paper was supported in part by grants 1367/2016 and 1161/2011 from the German–Israeli Science Foundation (GIF), Israel .
| Funders | Funder number |
|---|---|
| Blavatnik Research Fund in Computer Science | |
| German–Israeli Science Foundation | |
| ISAAC | 50:1–50:12 |
| ISF, Israel | 892/13, 260/18 |
| National Science Foundation | CCF-1421231, CCF-1217462 |
| Horizon 2020 Framework Programme | 757609 |
| Blavatnik Family Foundation | 1367/2016, 1161/2011 |
| European Commission | |
| Deutsche Forschungsgemeinschaft | MU/3501/1 |
| Tel Aviv University | 1595-19 |
| Israeli Centers for Research Excellence | 4/11 |
Keywords
- Disk intersection graph
- LP-type problem
- Stabbing set