Embeddability of Arrangements of Pseudocircles and Graphs on Surfaces

Abstract

A pseudocircle is a simple closed curve on some surface; an arrangement of pseudocircles is a collection of pseudocircles that pairwise intersect in exactly two points, at which they cross. Ortner proved that an arrangement of pseudocircles is embeddable into the sphere if and only if all of its subarrangements of size at most four are embeddable into the sphere, and asked if an analogous result holds for embeddability into orientable surfaces of higher genus. We answer this question positively: An arrangement of pseudocircles is embeddable into an orientable surface of genus g if and only if all of its subarrangements of size at most \(4g+4\) are. Moreover, this bound is tight. We actually have similar results for a much general notion of arrangement, which we call an arrangement of graphs.