Testing the necklace condition for Shortest Tours and optimal factors in the plane

Abstract

A tour τ of a finite set P of points is a necklace-tour if there are disks with the points in P as centers such that two disks intersect if and only if their centers are adjacent in τ. It has been observed by Sanders that a necklace-tour is an optimal traveling salesman tour. In this paper, we present an algorithm that either reports that no necklace-tour exists or outputs a necklace-tour of a given set of n points in O(n2logn) time. If a tour is given, then we can test in O(n2) time whether or not this tour is a necklace tour. Both algorithms can be generalized to m-factors of point sets in the plane. The complexity results rely on a combinatorial analysis of certain intersection graphs of disks defined for finite sets of points in the plane.