Type: Article

Embeddability of Arrangements of Pseudocircles and Graphs on Surfaces

Colin de Verdiere, Eric; Medina, Carolina; Roldan-Pensado, Edgardo; et ál..;

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 gif and only if all of its subarrangements of size at most4g+4 are. Moreover, this bound is tight. We actually have similar results for a much general notion of arrangement, which we call anarrangement of graphs.