Simple linear time recognition of unit interval graphs

We present a linear time algorithm for unit interval graph recognition. The algorithm is simple and based on Breadth-First Search. It is also direct — it does not first recognize the graph as an interval graph. Given a graph G, the algorithm produces an ordering of the vertices of the graph whenever G is a unit interval graph. This order corresponds to the order of the intervals of some unit interval model for G when arranged according to the increasing order of their left end coordinates. Breadth-First Search can also be used to construct a unit interval model for a unit interval graph on n vertices; in this model each endpoint is rational, with denominator n.