Decidability of string graphs

Abstract

We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a minimal drawing realizing a string graph in the plane. This upper bound confirms a conjecture by Kratochvı́l and Matoušek (J. Combin. Theory. Ser. B 53 (1991).) and settles the long-standing open problem of the decidability of string graph recognition (Bell System Tech. J. 45 (1996) 1639; Open Problem at Fifth Hungarian Collogium on Combinatories, 1976). Finally we show how to apply the result to solve another old open problem: deciding the existence of Euler diagrams, a fundamental problem of topological inference (Proceedings of the 14th International Joint Conference on Artificial Intelligence, 1995, p. 901). The general theory of Euler diagrams turns out to be as hard as second-order arithmetic.