Unit Interval Orders of Open and Closed Intervals

A poset P=(V,≺) is a unit OC interval order if there exists a representation that assigns an open or closed real interval I(x) of unit length to each x ∈ P so that x ≺ y in P precisely when each point of I(x) is less than each point in I(y). In this paper we give a forbidden poset characterization of the class of unit OC interval orders and an efficient algorithm for recognizing the class. The algorithm takes a poset P as input and either produces a representation or returns a forbidden poset induced in P

Interval Orders with Restrictions on the Interval Lengths

This thesis examines several classes of interval orders arising from restrictions on the permissible interval lengths. We first provide an accessible proof of the characterization theorem for the class of interval orders representable with lengths between 1 and k for each k in {1,2,...}. We then consider the interval orders representable with lengths exactly 1 and k for k in {0,1,...}. We characterize the class of interval orders representable with lengths 0 and 1, both structurally and algorithmically. To study the other classes in this family, we consider a related problem, in which each interval has a prescribed length. We derive a necessary and sufficient condition for an interval order to have a representation with a given set of prescribed lengths. Using this result, we provide a necessary condition for an interval order to have a representation with lengths 1 and 2