The Dimension of Finite and Infinite Comparability Graphs

Let Kbe a set, finite or infinite. A simple directed graph P = {V, U) is said to be "transitively oriented" if the existence of a directed path from a vertex x to a vertex y implies the existence of an arc (x, y)e U. For undefined graph-theoretical terms see for instance [4]. A transitively oriented graph on the vertex set V is nothing but the graph of some partial ordering of the set V; loops are omitted. Hence the subsequent denomination of pograph. An undirected graph G = (V, E) is called a comparability graph if it is possible to direct all edges in such a way that the resulting digraph P = {V, U) is transitively oriented. When such a relationship prevails between an undirected graph G and a directed graph P, we shall write G = <g(P). Notice that G = %P) = ^(P") , where P~ denotes, as usual, the partial ordering obtained by reversing each arc of P. If P and Q are pographs such that ^(P) = ^{Q) then it is not always the case that P = Q or P " 1 = Q. A comparability graph G is called UPO (short for uniquely partially orderable) if <g{P) = $(Q) = G implies that P = Qor P = Q~.