Abstract. We show that recognizing intersection graphs of convex sets has the same complexity as deciding truth in the existential theory of the reals. Comparing this to similar results on the rectilinear crossing number and intersection graphs of line segments, we argue that there is a need to recognize this level of complexity as its own class.

We introduce the complexity class ∃R based on the existential theory of the reals. We show that the definition of ∃R is robust in the sense that even the fragment of the theory expressing solvability of systems of strict polynomial inequalities leads to the same complexity class. Several natural and well-known problems turn out to be complete for ∃R; here we show that the complexity of decision variants of fixed-point problems, including Nash equilibria, are complete for this class, complementing work by Etessami and Yannakakis [13].

The crossing number is a popular tool in graph drawing and visualization, but there is not really just one crossing number; there is a large family of crossing number notions of which the crossing number is the best known. We survey the rich variety of crossing number variants that have been introduced in the literature for purposes that range from studying the theoretical underpinnings of the crossing number to crossing minimization for visualization problems.

We study Hanani-Tutte style theorems for various notions of planarity, including partially embedded planarity, and simultaneous planarity. This approach brings together the combinatorial, computational and algebraic aspects of planarity notions and may serve as a uniform foundation for planarity, as suggested in the writings of Tutte and Wu. 1 There were precursors to his approach, notably the paper by Hanani [12], but also work by Flores, van Kampen, and Wu. Some of the history can be found in [33].

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