Recognition and characterization of unit interval graphs with integer endpoints

We study those unit interval graphs having a model with intervals of integer endpoints and prescribed length. We present a structural result for this graph subclass which leads to a quadratic-time recognition algorithm, giving as positive certificate a model of minimum total length and as negative certificate a forbidden induced subgraph. We also present a quadratic-time algorithm to build, given a unit interval graph, a unit interval model with integer endpoints for which the interval length is as minimum as possible.