Recognizing Visibility Graphs of Triangulated Irregular Networks

Article type: Research Article

Authors: Boomari, Hossein^{; *} | Ostovari, Mojtaba | Zarei, Alireza

Affiliations: Department of Mathematical Science, Sharif University of Technology, Azadi Street, Tehran, Iran. h.boomari1@student.sharif.ir, m.ostovari@student.sharif.ir, zarei@sharif.ir

Correspondence: [*] Address for correspondence: Sharif University of Technology, Azadi Street, Tehran, Iran.

Abstract: A Triangulated Irregular Network (TIN) is a data structure that is usually used for representing and storing monotone geographic surfaces, approximately. In this representation, the surface is approximated by a set of triangular faces whose projection on the XY-plane is a triangulation. The visibility graph of a TIN is a graph whose vertices correspond to the vertices of the TIN and there is an edge between two vertices if their corresponding vertices on TIN see each other, i.e. the segment that connects these vertices completely lies above the TIN. Computing the visibility graph of a TIN and its properties have been considered thoroughly in the literature. In this paper, we consider this problem in reverse: Given a graph G, is there a TIN with the same visibility graph as G? We show that this problem is ∃ℝ-Complete.

Keywords: Visibility graph triangulated irregular network recognizing visibility graph existential theory of the reals

DOI: 10.3233/FI-2021-2027

Journal: Fundamenta Informaticae, vol. 179, no. 4, pp. 345-360, 2021