Bounded, minimal, and short representations of unit interval and unit circular-arc graphs

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 $\cal M$ is given and the goal is to obtain an equivalent UCA model $\cal U$. We show a linear time algorithm with negative certification that can also be implemented to run in logspace. In the bounded version, $\cal M$ is given together with some lower and upper bounds that the beginning points of $\cal U$ must satisfy. We develop a linear space $O(n^2)$ time algorithm for this problem. Finally, in the minimal version, the circumference of the circle and the length of the arcs in $\cal U$ must be simultaneously as minimum as possible. We prove that every UCA graph admits such a minimal model, and give a polynomial time algorithm to find it. We also consider the minimal representation problem for UIG graphs. As a bad result, we show that the previous linear time algorithm fails to provide a minimal model for some input graphs. We fix this algorithm but, unfortunately, it runs in linear space $O(n^2)$ time. Finally, we apply the minimal representation algorithms so as to find the minimum powers of paths and cycles that contain a given UIG and UCA models, respectively.