Almost all comparability graphs are UPO

Abstract

An undirected graph G is called a comparability graph if there exists an orientation of its edges such that the resulting relation on its vertex set is a partial order P. A comparability graph is UPO (i.e. uniquely partially orderable) if, except for its dual P −1, there is only one such partial order P. In this paper we show that lim n→∞ G(n, UPO) G(n) = 1 , where G( n) and G( n, UPO) denote, respectively, the number of comparability graphs and UPO-comparability graphs on n vertices. As a consequence, G( n) is asymptotically equal to half the number of partial orders on n elements.