Balanced Line Separators of Unit Disk Graphs

@inproceedings{Carmi2017BalancedLS,
  title={Balanced Line Separators of Unit Disk Graphs},
  author={Paz Carmi and Man-Kwun Chiu and Matthew J. Katz and Matias Korman and Yoshio Okamoto and Andr{\'e} van Renssen and Marcel Roeloffzen and Taichi Shiitada and Shakhar Smorodinsky},
  booktitle={WADS},
  year={2017}
}
We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of $n$ unit disks in the plane there exists a line $\ell$ such that $\ell$ intersects at most $O(\sqrt{(m+n)\log{n}})$ disks and each of the halfplanes determined by $\ell$ contains at most $2n/3$ unit disks from the set, where $m$ is the number of intersecting pairs of disks. We also show that an axis-parallel line intersecting $O(\sqrt{m+n})$ disks exists, but each halfplane may… 
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