Minimal and minimum unit circular-arc models

Abstract

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 $\cal U = (C,\cal A)$ is a $(c, \ell)$-CA model when $C$ has circumference $c$, all the arcs in $\cal A$ have length $\ell$, and all the extremes of the arcs in $\cal A$ are at a distance at least $1$. If $c \leq c'$ and $\ell \leq \ell'$ for every $(c', \ell')$-CA model equivalent (resp. isomorphic) to $\cal U$, then $\cal U$ is minimal (resp. minimum). In this article we prove that every PCA model is isomorphic to a minimum model. Our main tool is a new characterization of those PCA models that are equivalent to $(c,\ell)$-CA models, that allows us to conclude that $c$ and $\ell$ are integer when $\cal U$ is minimal. As a consequence, we obtain an $O(n^3)$ time and $O(n^2)$ space algorithm to solve the minimal representation problem, while we prove that the minimum representation problem is NP-complete.