Approximating Minimum Dominating Set on String Graphs

A string graph is an intersection graph of simple curves on the plane. For k >= 0, B-k-VPG graphs are intersection graphs of simple rectilinear curves having at most k cusps (bends). It is well-known that any string graph is a B-k-VPG graph for some value of k. For k >= 0, unit B-k-VPG graphs are intersection graphs of simple rectilinear curves having at most k cusps (bends) and each segment of the curve being unit length. Any string graph is a unit-B-k-VPG graph for some value of k. In this article, we show that the Minimum Dominating Set (MDS) problem for unit B-k-VPG graphs is NP-Hard for all k >= 1 and provide an O(k(4))-approximation algorithm for all k >= 0. Furthermore, we also provide an 8-approximation for the MDS problem for the vertically-stabbed L-graphs, intersection graphs of L-paths intersecting a common vertical line. The same problem is known to be APX-Hard (MFCS, 2018). As a by-product of our proof, we obtained a 2-approximation algorithm for the stabbing segment with rays (SSR) problem introduced and studied by Katz et al. (Comput. Geom. 2005).