Unit Interval Orders of Open and Closed Intervals

Abstract

A poset P = (X, ≺) 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 . arXiv:1501.06430v1 [math.CO]

Completion of the mixed unit interval graphs hierarchy

“…The proof of this result comes from Algorithm 17 in [8] that we used in the previous proof. This Algorithm takes a graph G supposed to be in U and gives a U-interval representation of it.…”

“…Checking that the interval representation is correct can be done in O(n + m), hence our result. [8]. However, it is possible to modify Algorithm 17 so as to recognize the class…”

“…However, this is not true for unit interval graphs: deciding which types of intervals are allowed to represent the vertices of a graph is crucial. This fact was notably studied, chronogically, [1], [2], [3], [6], [7] and [8]. In these papers one can find results about the classes of graphs we can get depending on the types of unit intervals we allow for their representations.…”

“…This class is a proper subclass of mixed unit interval graphs which are the graphs obtained when the intervals are only required to be of unit length. Recently, Joos [3] gave a characterization of mixed unit interval graphs by an infinite class of forbidden induced subgraphs, and Shuchat, Shull, Trenk and West [8] complemented it by a quadratic-time algorithm which, given any graph in this class, outputs a corresponding mixed unit interval reprensentation. In [5], Le and Rautenbach took a different approach and studied the graphs which are representable by intervals beginning at integer positions.…”

Tolerance Orders of Open and Closed Unit Intervals

“…These were first introduced in [9] in graph form and subsequently studied by other authors, e.g., [2,7,8,Type Interval Endpoints Center Table 1: The four types of intervals in an ABCD-representation. 13,14]. In this paper, we combine the concepts of 50% tolerance orders and unit interval orders by labeling the center points in one of two ways, called open and closed.…”