Bounded, minimal, and short representations of unit interval and unit circular-arc graphs. Chapter I: theory

Abstract

This is the first of two chapters of a work in which we consider the unrestricted, minimal, and bounded representation problems for unit interval (UIG) and unit circular-arc (UCA) graphs. In the unrestricted version, a proper circular-arc (PCA) model ${\mathcal{M}}$ is given and the goal is to obtain an equivalent UCA model ${\mathcal{U}}$. In the bounded version, ${\mathcal{M}}$ is given together with some lower and upper bounds that the beginning points of ${\mathcal{U}}$ must satisfy. In the minimal version, we have to find a minimal model equivalent to ${\mathcal{M}}$, in which the circumference of the circle and length of the arcs must be simultaneously as small as possible. In this chapter we motivate these problems from an historical perspective, and we develop the theoretical framework required for the algorithms in Chapter II. We present new characterizations of those PCA models that have equivalent UCA models, and of those UCA models with a circle of circumference ${c}$ and the arcs of length $\ell$. We also prove that every UCA model is equivalent to a minimal one. We remark that all our results are of an algorithmic nature and can be readily employed to solve the problems at hand, even though these algorithms are not as efficient as those in Chapter II