Non-crossing Connectors in the Plane
We consider the non-crossing connectors problem, which is stated as follows: Given n 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 ≠ 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 Pi consists of only two points on the boundary of Ri 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.
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