The Dimension of Finite and Infinite Comparability Graphs

1. Basic definitions and aim of the paper 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" 1), where P~ l 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~ l. Examples. An odd cycle is not a comparability graph, with the exception of the triangle. Every complete graph K n {n ^ 3) is a non-UPO comparability graph. Every connected bipartite graph is a UPO comparability graph. Comparability graphs have been studied by a large number of authors. Many of these worked independently and brought different insights to the problem but the central concept in all these works is that of forcing class, a notion first introduced by Gilmore and Hoffman [8], and Ghouila-Houri [7]. For a set C of edges of a graph G = (V, E) let V(C) be the set of vertices incident with some eeC. For edges e, e' e E we set e F e' if e = [x, y], e' = [x, z] and y, z are not joined in G. Since there are no loops, e F e for e e E. The binary relation F on the edge set E has a transitive closure F which is an equivalence relation on E. The equivalence classes …