Integrality of minimal unit circular-arc models

A proper circular-arc (PCA) model is a pair ${\cal M} = (C, {\cal A})$ where $C$ is a circle and $\cal A$ is a family of inclusion-free arcs on $C$ in which no two arcs of $\cal A$ cover $C$. A PCA model $\cal U$ is a $(c, \ell, d, d_s)$-CA model when $C$ has circumference $c$, all the arcs in $\cal A$ have length $\ell$, all the extremes of the arcs in $\cal A$ are at a distance at least $d$, and all the beginning points of the arcs in $\cal A$ are at a distance at least $d + d_s$. If $c \leq c'$ and $\ell \leq \ell'$ for every $(c', \ell', d, d_s)$-CA model, then $\cal U$ is $(d, d_s)$-minimal. In this article we prove that $c$ and $\ell$ are integer combinations of $d$ and $d_s$ when $\cal U$ is $(d, d_s)$-minimal. As a consequence, we obtain an algorithm to compute a $(d, d_s)$-minimal model equivalent to an input PCA model $\cal M$ that runs in $O(n^3)$ time and $O(n^2)$ space.