Non-crossing Connectors in the Plane

TL;DR

This paper investigates the non-crossing connectors problem in the plane, proving existence for pseudo-disk regions, providing a polynomial-time solution for axis-aligned rectangles, and establishing NP-completeness in the general case.

Contribution

It proves the existence of non-crossing connectors for pseudo-disk regions, offers an efficient algorithm for rectangles, and shows NP-completeness for the general problem.

Findings

Non-crossing connectors always exist for pseudo-disk regions.

A simple polynomial-time algorithm is available for axis-aligned rectangles.

The general problem is NP-complete even under restricted conditions.

Abstract

We consider the non-crossing connectors problem, which is stated as follows: Given n simply connected regions R_1,...,R_n in the plane and finite point sets P_i subset of R_i for i=1,...,n, are there non-crossing connectors y_i for (R_i,P_i), i.e., arc-connected sets y_i with P_i subset of y_i subset of R_i for every i=1,...,n, such that y_i and y_j are disjoint for all i different from 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 P_i consists of only two points on the boundary of R_i for i=1,...,n.