On dominating set of some subclasses of string graphs

We provide constant factor approximation algorithms for the Minimum Dominating Set (MDS) problem on several subclasses of string graphs i.e. intersection graphs of simple curves on the plane. For k ≥ 0, unit Bk-VPG graphs are intersection graphs of simple rectilinear curves having at most k cusps (bends) and each segment of the curve being unit length. We give an 18-approximation algorithm for the MDS problem on unit B0-VPG graphs. This partially addresses a question of Katz et al. (2005) [24]. We also give an O(k4)- approximation algorithm for the MDS problem on unit Bk-VPG graphs. We show that there is an 8-approximation algorithm for the MDS problem on vertically-stabbed L-graphs. We also give a 656-approximation algorithm for the MDS problem on stabbed rectangle overlap graphs. This is the first constant-factor approximation algorithm for the MDS problem on stabbed rectangle overlap graphs and extends a result of Bandyapadhyay et al. (2019) [31]. We prove some hardness results to complement the above results.