Linear Extension Diameter of Downset Lattices of 2-Dimensional Posets

The linear extension diameter of a finite poset P is the maximum distance between a pair of linear extensions of P, where the distance between two linear extensions is the number of pairs of elements of P appearing in different orders in the two linear extensions. We prove a formula for the linear extension diameter of the Boolean Lattice and characterize the diametral pairs of linear extensions. For the more general case of a downset lattice D_P of a 2-dimensional poset P, we characterize the diametral pairs of linear extensions of D_P and show how to compute the linear extension diameter of D_P in time polynomial in |P|.Comment: 25 pages, 7 figure

am Fachbereich Mathematik der

Die selbständige und eigenhändige Anfertigung versichere ich an Eide

Ideas for a Thesis Project: Drawing Partial Orders

Suppose we are given a partially ordered set P = (X, <) and aim to draw it in the plane. How can we find a good way of doing this? First of all: What makes a drawing good? One possible answer to this is that the structure of the poset should be clearly visible, another could be that the occupied area (or square area) should be minimized. If the given poset belongs to the subclass of 2-dimensional posets, good ways (in either sense) of drawing it are known. The dimension of a poset P is the minimum number of linear extensions whose intersection is P (see [1]). Thus, for each 2-dimensional poset P = (X, <) exists a pair L1, L2 of linear extensions in which every incomparable pair of P appears in both orders. Now there is an immediate way of representing P as point set in the plane: The coordinates of x ∈ X are determined by its position in L1 and L2. For y ∈ X we have x < y if and only if y is upward and to the right of x. This yields an embedding of P in a grid of size |X | × |X|. It can be turned into a drawing of P which nicely shows the structure of P. Figure 1 illustrates these well-known methods

Diametral pairs of linear extensions

Given a finite poset $\mathcal{P}$, we consider pairs of linear extensions of $\mathcal{P}$ with maximal distance, where the distance between two linear extensions $L_1, L_2$ is the number of pairs of elements of $\mathcal{P}$ appearing in different orders in $L_1$ and $L_2$. A diametral pair maximizes the distance among all pairs of linear extensions of $\mathcal{P}$. Felsner and Reuter defined the linear extension diameter of $\mathcal{P}$ as the distance between a diametral pair of linear extensions. We show that computing the linear extension diameter is NP-complete in general but can be solved in polynomial time for posets of width 3. Felsner and Reuter conjectured that, in every diametral pair, at least one of the linear extensions reverses a critical pair. We construct a counterexample to this conjecture. On the other hand, we show that a slightly stronger property holds for many classes of posets: we call a poset diametrally reversing if, in every diametral pair, both linear extensions reverse a critical pair. Among other results we show that interval orders and 3-layer posets are diametrally reversing. From the latter it follows that almost all posets are diametrally reversing