# Terrain-Like Graphs: PTASs for Guarding Weakly-Visible Polygons and Terrains

@inproceedings{Ashur2019TerrainLikeGP, title={Terrain-Like Graphs: PTASs for Guarding Weakly-Visible Polygons and Terrains}, author={Stav Ashur and Omrit Filtser and Matthew J. Katz and Rachel Saban}, booktitle={WAOA}, year={2019} }

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…

## 6 Citations

### Terrain-like Graphs and the Median Genocchi Numbers

- Mathematics
- 2022

A graph with vertex set { 1 , . . . , n } is terrain-like if, for any edge pair { a, c } , { b, d } with a < b < c < d , the edge { a, d } also exists. Terrain-like graphs frequently appear in…

### A Fast Shortest Path Algorithm on Terrain-like Graphs

- Computer Science, MathematicsDiscret. Comput. Geom.
- 2021

This paper devise a fast output-sensitive shortest path algorithm on a superclass of terrain visibility graphs called terrain-like graphs (including all induced subgraphs of terrain visible graphs), which runs in O ( d ∗ log Δ ) time.

### Advancing Through Terrains

- Mathematics, Computer ScienceArXiv
- 2019

A fast output-sensitive shortest path algorithm on terrain-like graphs and a polynomial-time algorithm for \textsc{Dominating Set} on special terrain visibility graphs (called funnel visibility graphs) are devised.

### Guarding Weakly-Visible Polygons with Half-Guards

- Mathematics, Computer Science
- 2022

A polynomial time approximation scheme for vertex guarding a weakly-visible polygon with half-guards is given and it is shown that even with many restrictions, the problem is NP-hardness.

### A Constant-Factor Approximation Algorithm for Point Guarding an Art Gallery

- Computer ScienceArXiv
- 2021

This paper proposes an algorithm with a constant approximation factor for the point guarding problem where the location of guards is restricted to a grid and the running time depends on the number of cells of the grid.

### Art Gallery Plus Single Specular-reflection

- Computer ScienceArXiv
- 2021

A variant of the Art Gallery problem in which the boundaries of P are replaced by single specularreflection edges, allowing the view rays to reflect once per collision with an edge, which allows the guards to see through the reflections, thereby viewing a larger portion of the polygon is studied.

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