License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2020.6
URN: urn:nbn:de:0030-drops-121640
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12164/
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Ameer, Safwa ; Gibson-Lopez, Matt ; Krohn, Erik ; Soderman, Sean ; Wang, Qing

Terrain Visibility Graphs: Persistence Is Not Enough

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LIPIcs-SoCG-2020-6.pdf (0.5 MB)


Abstract

In this paper, we consider the Visibility Graph Recognition and Reconstruction problems in the context of terrains. Here, we are given a graph G with labeled vertices v₀, v₁, …, v_{n-1} such that the labeling corresponds with a Hamiltonian path H. G also may contain other edges. We are interested in determining if there is a terrain T with vertices p₀, p₁, …, p_{n-1} such that G is the visibility graph of T and the boundary of T corresponds with H. G is said to be persistent if and only if it satisfies the so-called X-property and Bar-property. It is known that every "pseudo-terrain" has a persistent visibility graph and that every persistent graph is the visibility graph for some pseudo-terrain. The connection is not as clear for (geometric) terrains. It is known that the visibility graph of any terrain T is persistent, but it has been unclear whether every persistent graph G has a terrain T such that G is the visibility graph of T. There actually have been several papers that claim this to be the case (although no formal proof has ever been published), and recent works made steps towards building a terrain reconstruction algorithm for any persistent graph. In this paper, we show that there exists a persistent graph G that is not the visibility graph for any terrain T. This means persistence is not enough by itself to characterize the visibility graphs of terrains, and implies that pseudo-terrains are not stretchable.

BibTeX - Entry

@InProceedings{ameer_et_al:LIPIcs:2020:12164,
  author =	{Safwa Ameer and Matt Gibson-Lopez and Erik Krohn and Sean Soderman and Qing Wang},
  title =	{{Terrain Visibility Graphs: Persistence Is Not Enough}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{6:1--6:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Sergio Cabello and Danny Z. Chen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12164},
  URN =		{urn:nbn:de:0030-drops-121640},
  doi =		{10.4230/LIPIcs.SoCG.2020.6},
  annote =	{Keywords: Terrains, Visibility Graph Characterization, Visibility Graph Recognition}
}

Keywords: Terrains, Visibility Graph Characterization, Visibility Graph Recognition
Collection: 36th International Symposium on Computational Geometry (SoCG 2020)
Issue Date: 2020
Date of publication: 08.06.2020


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