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 $$4g+4$$ 4 g + 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 .

中文翻译：

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表面上伪圆和图形排列的可嵌入性

伪圆是某个曲面上的简单闭合曲线；伪圆的排列是在两个点上成对相交的一组伪圆，它们在这两个点上相交。Ortner 证明了伪圆的排列可嵌入到球体中，当且仅当其大小最多为 4 的所有子排列都可嵌入到球体中，并询问类似的结果是否适用于可嵌入更高属的可定向表面。我们肯定地回答这个问题：当且仅当其所有大小至多 $$4g+4$$4g+4 的子排列都可以嵌入到 g 属的可定向表面中。而且，这个界限很紧。对于一个非常普遍的排列概念，我们实际上有类似的结果，我们称之为图的排列。