Recognizing Visibility Graphs of Triangulated Irregular Networks

TL;DR

This paper investigates whether a given graph can be realized as the visibility graph of a Triangulated Irregular Network (TIN), establishing the problem's computational complexity as complete for the existential theory of the reals.

Contribution

It introduces the problem of recognizing TIN visibility graphs from arbitrary graphs and proves its computational complexity class.

Findings

The recognition problem is complete for the existential theory of the reals.

Provides a formal complexity classification for TIN visibility graph recognition.

Addresses the reverse problem of computing visibility graphs of TINs.

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 Complete for Existential Theory of The Reals.