Nesting points in the sphere (Q1349071)

Let \(G\) be a graph imbedded in the sphere \(S\), with \(x\) a point of \(S-G\). A \(k\)-nest of \(x\) is a collection of cycles \(C_1,\dots, C_k\) of \(G\) so that for each \(i\), the side of \(C_i\) containing \(x\) also contains \(C_j\) for all \(j< i\). Then \(G\) is \(k\)-nested if every point \(x\) in \(S-G\) has a \(k\)-nest. The authors find the minor-minimal \(k\)-nested spherical maps for small \(k\). In particular, they find the obstructions (under the minor order) for the class of planar maps having a face whose boundary meets all other face boundaries.