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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 adjacentwhenever the two corresponding points are visible from each other, that is, theopen segment between them does not contain any other point of the set. We studythe recognition problem for point visibility graphs: given a simple undirectedgraph, decide whether it is the visibility graph of some point set in theplane. We show that the problem is complete for the existential theory of thereals. Hence the problem is as hard as deciding the existence of a realsolution to a system of polynomial inequalities. The proof involves simplesubstructures forcing collinearities in all realizations of some visibilitygraphs, which are applied to the algebraic universality constructions of Mn\"evand Richter-Gebert. This solves a longstanding open question and paves the wayfor the analysis of other classes of visibility graphs. Furthermore, as acorollary of one of our construction, we show that there exist point visibilitygraphs that do not admit any geometric realization with points having integercoordinates.

Recognition and Complexity of Point Visibility Graphs