Non-crossing Connectors in the Plane

We consider the non-crossing connectors problem, which is stated as follows: Given n simply connected regions R1, . . . , Rn in the plane and finite point sets Pi ⊂ Ri for i = 1, . . . , n, are there non-crossing connectors γi for (Ri, Pi), i.e., arc-connected sets γi with Pi ⊂ γi ⊂ Ri for every i = 1, . . . , n, such that γi ∩ γj = ∅ for all i 6= j? We prove that non-crossing connectors do always exist if the regions form a collection of pseudo-disks, i.e., the boundaries of every pair of regions intersect at most twice. We provide a simple polynomial-time algorithm if the regions are axis-aligned rectangles. Finally we prove that the general problem is NP-complete, even if the regions are convex, the boundaries of every pair of regions intersect at most four times and Pi consists of only two points on the boundary of Ri for i = 1, . . . , n.