Integrality of minimal unit circular-arc models

A proper circular-arc (PCA) model is a pairM = (C,A) where C is a circle and A is a family of inclusion-free arcs on C in which no two arcs of A cover C. A PCA model U is a (c, `, d, ds)-CA model when C has circumference c, all the arcs in A have length `, all the extremes of the arcs in A are at a distance at least d, and all the beginning points of the arcs in A are at a distance at least d+ ds. If c ≤ c′ and ` ≤ `′ for every (c′, `′, d, ds)-CA model, then U is (d, ds)-minimal. In this article we prove that c and ` are integer combinations of d and ds when U is (d, ds)-minimal. As a consequence, we obtain an algorithm to compute a (d, ds)-minimal model equivalent to an input PCA modelM that runs in O(n3) time and O(n2) space.