### Abstract

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 ℓ such that ℓ intersects at most O((m+n)logn) disks and each of the halfplanes determined by ℓ 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(m+n) disks exists, but each halfplane may contain up to 4n/5 disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in n exists when we look at disks of arbitrary radii, even when m=0. Proofs are constructive and suggest simple algorithms that run in linear time. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size O(m)).

Original language | English |
---|---|

Article number | 101575 |

Number of pages | 14 |

Journal | Computational Geometry: Theory and Applications |

Volume | 86 |

DOIs | |

Publication status | Published - 1 Jan 2020 |

### Fingerprint

### Keywords

- Balanced separator
- Centerpoint
- Geometric intersection graph
- Line separator
- Unit disk graph

### Cite this

*Computational Geometry: Theory and Applications*,

*86*, [101575]. https://doi.org/10.1016/j.comgeo.2019.101575

}

*Computational Geometry: Theory and Applications*, vol. 86, 101575. https://doi.org/10.1016/j.comgeo.2019.101575

**Balanced line separators of unit disk graphs.** / Carmi, Paz; Chiu, Man Kwun; Katz, Matthew J.; Korman, Matias; Okamoto, Yoshio; van Renssen, André; Roeloffzen, Marcel; Shiitada, Taichi (Corresponding author); Smorodinsky, Shakhar.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Balanced line separators of unit disk graphs

AU - Carmi, Paz

AU - Chiu, Man Kwun

AU - Katz, Matthew J.

AU - Korman, Matias

AU - Okamoto, Yoshio

AU - van Renssen, André

AU - Roeloffzen, Marcel

AU - Shiitada, Taichi

AU - Smorodinsky, Shakhar

PY - 2020/1/1

Y1 - 2020/1/1

N2 - 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 ℓ such that ℓ intersects at most O((m+n)logn) disks and each of the halfplanes determined by ℓ 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(m+n) disks exists, but each halfplane may contain up to 4n/5 disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in n exists when we look at disks of arbitrary radii, even when m=0. Proofs are constructive and suggest simple algorithms that run in linear time. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size O(m)).

AB - 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 ℓ such that ℓ intersects at most O((m+n)logn) disks and each of the halfplanes determined by ℓ 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(m+n) disks exists, but each halfplane may contain up to 4n/5 disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in n exists when we look at disks of arbitrary radii, even when m=0. Proofs are constructive and suggest simple algorithms that run in linear time. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size O(m)).

KW - Balanced separator

KW - Centerpoint

KW - Geometric intersection graph

KW - Line separator

KW - Unit disk graph

UR - http://www.scopus.com/inward/record.url?scp=85072551207&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2019.101575

DO - 10.1016/j.comgeo.2019.101575

M3 - Article

AN - SCOPUS:85072551207

VL - 86

JO - Computational Geometry

JF - Computational Geometry

SN - 0925-7721

M1 - 101575

ER -