Stabbing Pairwise Intersecting Disks by Four Points

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.