Unit Mixed Interval Graphs

In this paper we extend the work of Rautenbach and Szwarcfiter by giving a structural characterization of graphs that can be represented by the intersection of unit intervals that may or may not contain their endpoints. A characterization was proved independently by Joos, however our approach provides an algorithm that produces such a representation, as well as a forbidden graph characterization

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

The Total Weak Discrepancy of a Partially Ordered Set

We define the total weak discrepancy of a poset P as the minimum nonnegative integer k for which there exists a function f : V → Z satisfying (i) if a \prec b then f(a) + 1 ≤ f(b) and (ii) Σ|f(a) − f(b)| ≤ k, where the sum is taken over all unordered pairs {a, b} of incomparable elements. If we allow k and f to take real values, we call the minimum k the fractional total weak discrepancy of P. These concepts are related to the notions of weak and fractional weak discrepancy, where (ii) must hold not for the sum but for each individual pair of incomparable elements of P. We prove that, unlike the latter, the total weak and fractional total weak discrepancy of P are always the same, and we give a polynomial-time algorithm to find their common value. We use linear programming duality and complementary slackness to obtain this result

Range of the Fractional Weak Discrepancy Function

In this paper we describe the range of values that can be taken by the fractional weak discrepancy of a poset and characterize semiorders in terms of these values. In [6], we defined the fractional weak discrepancy wdF (P) of a poset P=(V,≺) to be the minimum nonnegative k for which there exists a function f:V→R satisfying (1) if a≺b then f(a)+1≤f(b) and (2) if a∥b then |f(a)−f(b)|≤k. This notion builds on previous work on weak discrepancy in [3, 7, 8]. We prove here that the range of values of the function wdF is the set of rational numbers that are either at least one or equal to r [over] r+1 for some nonnegative integer r. Moreover, P is a semiorder if and only if wdF (P) \u3c 1, and the range taken over all semiorders is the set of such fractions r [over] r+1

Interval Digraphs and Bounded Bitolerance Digraphs

Interval digraphs and bounded bitolerance digraphs are two different directed graph analogues to the well-known interval graphs. Both are contained in the class of digraphs of Ferrers Dimension at most two. In this paper we explore the relationship between these two important classes of digraphs

Fractional weak discrepancy and interval orders

The fractional weak discrepancy wdF (P) of a poset P = (V, ≺) was introduced in [6] as the minimum nonnegative k for which there exists a function f: V → R satisfying (i) if a ≺ b then f(a)+1 ≤ f(b) and (ii) if a � b then |f(a) − f(b) | ≤ k. In this paper we generalize results in [7, 8] on the range of the wdF function for semiorders (interval orders with no induced 3 + 1) to interval orders with no n + 1, where n ≥ 3. In particular, we prove that the range for such posets P is the set of rationals that can be written as r/s, where 0 ≤ s − 1 ≤ r < (n − 2)s. If wdF (P) = r/s and P has an optimal forcing cycle C with up(C) = r and side(C) = s, then r ≤ (n − 2)(s − 1). Moreover when s ≥ 2, for each r satisfying s − 1 ≤ r ≤ (n − 2)(s − 1) there is an interval order having such an optimal forcing cycle and containing no n + 1