Recognition and Complexity of Point Visibility Graphs

TL;DR

This paper investigates the computational complexity of recognizing point visibility graphs, proving it is as hard as solving polynomial inequalities over the reals, and shows some such graphs cannot be realized with integer coordinates.

Contribution

It establishes the recognition problem as complete for the existential theory of the reals and demonstrates the existence of non-integer realizable visibility graphs.

Findings

Recognition problem is complete for the existential theory of the reals.

Some point visibility graphs cannot be realized with integer coordinates.

The results solve a longstanding open problem in computational geometry.

Abstract

A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open segment between them does not contain any other point of the set. We study the recognition problem for point visibility graphs: given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals. Hence the problem is as hard as deciding the existence of a real solution to a system of polynomial inequalities. The proof involves simple substructures forcing collinearities in all realizations of some visibility graphs, which are applied to the algebraic universality constructions of Mn\"ev and Richter-Gebert. This solves a longstanding open question and paves the way for the analysis of other classes of visibility graphs. Furthermore, as a corollary of one of our construction, we show that there exist point visibility graphs that do not admit any geometric realization with points having integer coordinates.