Abstract
A string graph is an intersection graph of simple curves on the plane. For \(k\ge 0\), \(B_k\)-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 \(k\ge 0\), 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 \(k \ge 1\) and provide an \(O(k^4)\)-approximation algorithm for all \(k\ge 0\). 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).
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Chakraborty, D., Das, S., Mukherjee, J. (2019). Approximating Minimum Dominating Set on String Graphs. In: Sau, I., Thilikos, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2019. Lecture Notes in Computer Science(), vol 11789. Springer, Cham. https://doi.org/10.1007/978-3-030-30786-8_18
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