Abstract
A graph \(G = (V,E)\) is terrain-like if one can assign a unique integer from the range [1..|V|] to each vertex in V, such that, if both \(\{i,k\}\) and \(\{j,l\}\) are in E, for any \(i< j< k < l\), then so is \(\{i,l\}\). We present a local-search-based PTAS for minimum dominating set in terrain-like graphs. Then, we observe that, besides the visibility graphs of x-monotone terrains which are terrain-like, so are the visibility graphs of weakly-visible polygons and weakly-visible terrains, immediately implying a PTAS for guarding the vertices of such a polygon or terrain from its vertices. We also present PTASs for continuously guarding the boundary of a WV-polygon or WV-terrain, either from its vertices, or, for a WV-terrain, from arbitrary points on the terrain. Finally, we compare between terrain-like graphs and non-jumping graphs, and also observe that both families admit PTASs for maximum independent set.
M. J. Katz—Supported by grant 1884/16 from the Israel Science Foundation.
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Ashur, S., Filtser, O., Katz, M.J., Saban, R. (2020). Terrain-Like Graphs: PTASs for Guarding Weakly-Visible Polygons and Terrains. In: Bampis, E., Megow, N. (eds) Approximation and Online Algorithms. WAOA 2019. Lecture Notes in Computer Science(), vol 11926. Springer, Cham. https://doi.org/10.1007/978-3-030-39479-0_1
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