The Dimension of a Comparability Graph

Dushnik and Miller defined the dimension of a partial order P as the minimum number of linear orders whose intersection is P. Ken Bogart asked if the dimension of a partial order is an invariant of the associated comparability graph. In this paper we answer Bogart's question in the affirmative. The proof involves a characterization of the class of comparability graphs defined by Aigner and Prins as uniquely partially orderable graphs. Our characterization of uniquely partially orderable graphs is another instance of the frequently encountered phenomenon where the obvious necessary condition is also sufficient. 1. Notation and definitions. In this paper we consider a partial order as an irreflexive, transitive binary relation. With a binary relation R on a set A we associate a graph G(R) whose vertex set is A with distinct vertices x and y joined by an edge iffx R y ory R x. A graph G is called a comparability graph if there exists a partial order P for which G = G(P). Aigner and Prins [1] called a comparability graph G a uniquely partially orderable (UPO) graph if G = G(P) = G(Q) implies P = Q or P = Q where Q denotes the dual of Q. Let X be a graph and let {Gx I x E V(X )} be a family of graphs. Then the (Sabidussi) X-join [9] of this family is the graph with vertex set {(x,y)I x E V(X), y E V(G,)1 with (x, y) adjacent to (z, w) if x is adjacent to z in X or x = z and y is adjacent to w in Gx. Every graph X is isomorphic to the X-join of a family of trivial graphs. If a graph G is isomorphic to the X-join of a family {Gx I x E V(X )} where X is nontrivial and at least one Gx is nontrivial, then G is said to be decomposable; otherwise G is said to be indecomposable. Let G be a graph and let K be a subset of V(G). K is said to be partitive if for every vertex x with x M K, if there exists a vertex y E K such that x and y are adjacent, then x is adjacent to every vertex in K. A partitive subset K is said to be nontrivial when K is not the empty set, a singleton, or the entire vertex set. It is easy to see that a graph is indecomposable if it has no nontrivial partitive sets. Now let P be a partial order on a set A and let {Qala E A) be a family of partial orders. If we denote the set on which each Qa is defined by Aa, then the ordinal product [2] of this family over P is the partial order S on the set {(a,b)a A,b AEa} in which (al,b1)S(a2,b2) iff a Pa2 or a, = a2 and b, Qal b2. Clearly the comparability graph G(S) is the G(P)-join of the family {G(Qa)la E A). Let e andf be edges of a graph G. Gilmore and Hoffman [6] defined a strong Received by the editors May 12, 1975 and, in revised form, January 20, 1976. AMS (MOS) subject classifications (1970). Primary 06A10, 05C20.