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.
Highlights
Visibility between geometric objects is a cornerstone notion in discrete and computational geometry, that appeared as soon as the late 1960s in pioneering experiments in robotics [17]
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
A point visibility graph associated with a set P of points in the plane is a simple undirected graph G = (P, E) such that two points of P are adjacent if and only if the open segment between them does not
Summary
Visibility between geometric objects is a cornerstone notion in discrete and computational geometry, that appeared as soon as the late 1960s in pioneering experiments in robotics [17]. Despite decades of research on those topics, the combinatorial structures induced by visibility relations in the plane are far from understood. Among such structures, visibility graphs are arguably the most natural. A visibility graph encodes the binary, symmetric visibility relation among sets of objects in the plane, where two objects are visible from each other whenever there exists a straight line of sight between them that does not meet any obstacle. © Jean Cardinal and Udo Hoffmann; licensed under Creative Commons License CC-BY 31st International Symposium on Computational Geometry (SoCG’15). We will use the abbreviation PVG for point visibility graph
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