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The Dominating Set Problem in Geometric Intersection Graphs

Authors Mark de Berg, Sándor Kisfaludi-Bak, Gerhard Woeginger

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  • dominating set
  • intersection graph
  • W-hierarchy

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Abstract

We study the parameterized complexity of dominating sets 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. This generalizes known results from the literature. Finally, we establish W[1]-hardness for a large class of intersection graphs.

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Mark de Berg, Sándor Kisfaludi-Bak, and Gerhard Woeginger. The Dominating Set Problem in Geometric Intersection Graphs. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 14:1-14:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.IPEC.2017.14

Author Details

Mark de Berg
Sándor Kisfaludi-Bak
Gerhard Woeginger

References

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