Terrain-like graphs: PTASs for guarding weakly-visible polygons and terrains

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.