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Stabbing Pairwise Intersecting Disks by Four Points

In their seminal work, Danzer (1956, 1986) and Stachó (1981) established that every set of pairwise intersecting disks in the plane can be stabbed by four points. However, both these proofs are non-constructive, at least in the sense that they do not seem to imply an efficient algorithm for finding the stabbing points, given such a set of disks $D$. Recently, Har-Peled \etal (2018) presented a relatively simple linear-time algorithm for finding five points that stab $D$. We present an alternative proof (and the first in English) 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 liner-time algorithm with an elementary proof for finding five points that stab $D$.