The Dominating Set Problem in Geometric Intersection Graphs

TL;DR

This paper explores the parameterized complexity of the Dominating Set problem in various geometric intersection graphs, providing polynomial algorithms, NP-completeness results, and complexity class placements across different dimensions and graph types.

Contribution

It offers new complexity classifications for Dominating Set in geometric intersection graphs, including polynomial solvability, NP-completeness, and W[1] classifications in higher dimensions.

Findings

Polynomial solvability when Q contains an interval.

NP-completeness when Q has no intervals and irrational ratios.

W[1]-hardness results for certain classes of intersection graphs.

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.