Paradoxes of two-length interval orders

AbstractA two-length interval order is a partially ordered set whose points can be mapped into closed real intervals such that (i) the interval for x lies wholly to the right of the interval for y if and only if x is ranked above y in the partial ordering, and (ii) only two different lengths are involved in the mapping. With the shorter length fixed at 1, let L denote the set of admissible longer lengths for which (i) and (ii) hold for a given interval order.The paper demonstrates that there are two-length interval orders on finite point sets with the following L sets for each integer m⩾2: L = (1,m); L = (2−1m, 2)∪(m,∞); L = (m,2m− 1)∪(2m−1,∞). The second case shows that L can have an arbitrarily big gap between admissible longer lengths, and the third case leads to the corollary that there can be arbitrarily many gaps or holes in L

Interval graphs and interval orders

AbstractThis paper explores the intimate connection between finite interval graphs and interval orders. Special attention is given to the family of interval orders that agree with, or provide representations of, an interval graph. Two characterizations (one by P. Hanlon) of interval graphs with essentially unique agreeing interval orders are noted, and relationships between interval graphs and interval orders that concern the number of lengths required for interval representations and bounds on lengths of representing intervals are discussed.Two invariants of the family of interval orders that agree with an interval graph are established, namely magnitude, which affects end-point placements, and the property of having the lengths of all representing intervals between specified bounds. Extremization problems for interval graphs and interval orders are also considered

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