### Abstract

We study the parameterized complexity of the dominating set problem in geometric intersection graphs. • In one dimension, we investigate intersection graphs induced by translates of a fixed pattern Q that consists of a finite number of intervals and a finite number of isolated points. We prove that DOMINATING SET on such intersection graphs is polynomially solvable whenever Q contains at least one interval, and whenever Q contains no intervals and for any two point pairs in Q the distance ratio is rational. The remaining case where Q contains no intervals but does contain an irrational distance ratio is shown to be NP-complete and contained in FPT (when parameterized by the solution size).• In two and higher dimensions, we prove that DOMINATING SET is contained in W[1] for intersection graphs of semi-algebraic sets with constant description complexity. So far this was only known for unit squares. Finally, we establish W[1]-hardness for a large class of intersection graphs.

Original language | English |
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Pages (from-to) | 18-31 |

Number of pages | 14 |

Journal | Theoretical Computer Science |

Volume | 769 |

DOIs | |

Publication status | Published - 17 May 2019 |

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### Keywords

- Dominating Set
- Geometric intersection graph
- Semi-algebraic set
- W-hierarchy

### Cite this

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**The complexity of Dominating set in geometric intersection graphs.** / de Berg, Mark; Kisfaludi-Bak, Sándor (Corresponding author); Woeginger, Gerhard.

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

TY - JOUR

T1 - The complexity of Dominating set in geometric intersection graphs

AU - de Berg, Mark

AU - Kisfaludi-Bak, Sándor

AU - Woeginger, Gerhard

PY - 2019/5/17

Y1 - 2019/5/17

N2 - We study the parameterized complexity of the dominating set problem in geometric intersection graphs. • In one dimension, we investigate intersection graphs induced by translates of a fixed pattern Q that consists of a finite number of intervals and a finite number of isolated points. We prove that DOMINATING SET on such intersection graphs is polynomially solvable whenever Q contains at least one interval, and whenever Q contains no intervals and for any two point pairs in Q the distance ratio is rational. The remaining case where Q contains no intervals but does contain an irrational distance ratio is shown to be NP-complete and contained in FPT (when parameterized by the solution size).• In two and higher dimensions, we prove that DOMINATING SET is contained in W[1] for intersection graphs of semi-algebraic sets with constant description complexity. So far this was only known for unit squares. Finally, we establish W[1]-hardness for a large class of intersection graphs.

AB - We study the parameterized complexity of the dominating set problem in geometric intersection graphs. • In one dimension, we investigate intersection graphs induced by translates of a fixed pattern Q that consists of a finite number of intervals and a finite number of isolated points. We prove that DOMINATING SET on such intersection graphs is polynomially solvable whenever Q contains at least one interval, and whenever Q contains no intervals and for any two point pairs in Q the distance ratio is rational. The remaining case where Q contains no intervals but does contain an irrational distance ratio is shown to be NP-complete and contained in FPT (when parameterized by the solution size).• In two and higher dimensions, we prove that DOMINATING SET is contained in W[1] for intersection graphs of semi-algebraic sets with constant description complexity. So far this was only known for unit squares. Finally, we establish W[1]-hardness for a large class of intersection graphs.

KW - Dominating Set

KW - Geometric intersection graph

KW - Semi-algebraic set

KW - W-hierarchy

UR - http://www.scopus.com/inward/record.url?scp=85054885234&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2018.10.007

DO - 10.1016/j.tcs.2018.10.007

M3 - Article

AN - SCOPUS:85054885234

VL - 769

SP - 18

EP - 31

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -