How do curved spheres intersect in 3-space?

The following problem was proposed in 2010 by S. Lando. Let M and N be two unions of the same number of disjoint circles in a sphere. Do there always exist two spheres in 3-space such that their intersection is transversal and is a union of disjoint circles that is situated as M in one sphere and as N in the other? Union ^{M '} of disjoint circles is situated in one sphere as union M of disjoint circles in the other sphere if there is a homeomorphism between these two spheres which maps ^{M '} to M. We prove (by giving an explicit example) that the answer to this problem is "no". We also prove a necessary and sufficient condition on M and N for existing of such intersecting spheres. This result can be restated in terms of graphs. Such restatement allows for a trivial brute-force algorithm checking the condition for any given M and N. It is an open question if a faster algorithm exists.