Nesting Points in the Sphere

Let G be a graph embedded in the sphere. A k-nest of a point x not in G is a collection C 1 ; : : : ; C k of disjoint cycles such that for each C i , the side contain x also contains C j for each j ! i. An embedded graph is k-nested if each point not on the graph has a k-nest. In this paper we examine k-nested maps. We find the minor-minimal k-nested maps small values of k. In particular, we find the obstructions (under the minor order) for the class of planar maps with the property that one face's boundary meets all other face boundaries. 1 Permanent e-mail: [email protected] 1 1 Introduction Let G be a spherical graph, a graph drawn without crossings on the 2dimensional sphere S. Our interest is in separating some points in the sphere from others using simple cycles in G. Of course, this is not always possible: two points in the same component, or same face, of S \Gamma G are not separated. On the other hand, in some cases two points can be separated by many "..