Recognition and Complexity of Point Visibility Graphs

• Published in: Discrete & computational geometry Vol. 57; no. 1; pp. 164 - 178
• Main Authors: Cardinal, Jean, Hoffmann, Udo
• Format: Journal Article
• Language: English
• Published: New York Springer US 2017; Springer Nature B.V
• Subjects: Collinearity; Combinatorics; Computational Mathematics and Numerical Analysis; Geometry; Graphs; Inequalities; Mathematics; Mathematics and Statistics; Planes; Polynomials; Recognition; Segments; Visibility; Point visibility graphs; Geometric graph representations; Existential theory of the reals
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-016-9831-1

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 Mnëv 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 on a grid. We also prove that the problem of recognizing visibility graphs of points on a grid is decidable if and only if the existential theory of the rationals is decidable.