Non-crossing connectors in the plane

Abstract

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 = θ 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 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. © Springer-Verlag Berlin Heidelberg 2013.