Ideas for a Thesis Project: Drawing Partial Orders

Abstract

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