We consider the non-crossing connectors problem, which is stated as follows: Given n regions R ₁,…,R _n in the plane and finite point sets P _i  ⊂ R _i for i = 1,…,n, are there non-crossing connectors γ _i for (R _i ,P _i ), i.e., arc-connected sets γ _i with P _i  ⊂ γ _i  ⊂ R _i for every i = 1,…,n, such that γ _i  ∩ γ _j  = ∅ for all i ≠ 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 each region is the convex hull of the corresponding point set, or if all regions are axis-aligned rectangles. We prove that the general problem is NP-hard, even if the regions are convex, the boundaries of every pair of regions intersect at most four times and P _i consists of only two points on the boundary of R _i for i = 1,…,n.

Finally, we prove that the non-crossing connectors problem lies in NP, i.e., is NP-complete, by a reduction to a non-trivial problem, and that there indeed are problem instances in which every solution has exponential complexity, even when all regions are convex pseudo-disks.