Mitchel T. Keller

Posets with Cover Graph of Pathwidth two have Bounded Dimension

Joret, Micek, Milans, Trotter, Walczak, and Wang recently asked if there exists a constant d such that if P is a poset with cover graph of P of pathwidth at most 2, then dim(P)=d. We answer this question in the affirmative by showing that d=17 is sufficient. We also show that if P is a poset containing the standard example S5 as a subposet, then the cover graph of P has treewidth at least 3.

A Characterization of Partially Ordered Sets with Linear Discrepancy Equal to 2

The linear discrepancy of a poset P is the least k such that there is a linear extension L of P such that if x and y are incomparable in P, then |hL(x)–hL(y)|≤k, where hL(x) is the height of x in L. Tanenbaum, Trenk, and Fishburn characterized the posets of linear discrepancy 1 as the semiorders of width 2 and posed the problem of characterizing the posets of linear discrepancy 2. We show that this problem is equivalent to finding the posets with linear discrepancy equal to 3 having the property that the deletion of any point results in a reduction in the linear discrepancy. Howard determined that there are infinitely many such posets of width 2. We complete the forbidden subposet characterization of posets with linear discrepancy equal to 2 by finding the minimal posets of width 3 with linear discrepancy equal to 3. We do so by showing that, with a small number of exceptions, they can all be derived from the list for width 2 by the removal of specific comparisons.

Online Linear Discrepancy of Partially Ordered Sets

This article is dedicated to Professor Endre Szemeredi on the occasion of his 70th birthday. Among his many remarkable contributions to combinatorial mathematics and theoretical computer science is a jewel for online problems for partially ordered sets: the fact that h(h + l)/2 antichains are required for an online antichain partition of a poset of height h.

The Reversal Ratio of a Poset

Felsner and Reuter introduced the linear extension diameter of a partially ordered set P, denoted led(P), as the maximum distance between two linear extensions of P, where distance is defined to be the number of incomparable pairs appearing in opposite orders (reversed) in the linear extensions. In this paper, we introduce the reversal ratio RR(P) of P as the ratio of the linear extension diameter to the number of (unordered) incomparable pairs. We use probabilistic techniques to provide a family of posets Pk on at most k log k elements for which the reversal ratio RR(Pk) ≤ C / log k, where C is an absolute constant. We also examine the questions of bounding the reversal ratio in terms of order dimension and width.