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Terrain Visibility Graphs: Persistence Is Not Enough

Authors Safwa Ameer, Matt Gibson-Lopez , Erik Krohn, Sean Soderman, Qing Wang

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Author Details

Safwa Ameer
  • Department of Computer Science, The University of Texas at San Antonio, TX, USA
Matt Gibson-Lopez
  • Department of Computer Science, The University of Texas at San Antonio, TX, USA
Erik Krohn
  • Department of Computer Science, The University of Wisconsin, Oshkosh, WI, USA
Sean Soderman
  • Department of Computer Science, The University of Texas at San Antonio, TX, USA
Qing Wang
  • Department of Computer Science, The University of Tennessee at Martin, TN, USA

Funding

  • Gibson-Lopez, Matt: Supported by the National Science Foundation under Grant No. 1733874.
  • Soderman, Sean: Supported by the National Science Foundation under Grant No. 1733874.

Cite As Get BibTex

Safwa Ameer, Matt Gibson-Lopez, Erik Krohn, Sean Soderman, and Qing Wang. Terrain Visibility Graphs: Persistence Is Not Enough. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.6

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Terrains
  • Visibility Graph Characterization
  • Visibility Graph Recognition

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