On the shape of a random acyclic digraph

Abstract

AbstractIf D is an acyclic digraph, define the height h = h(D) to be the length of the longest directed path in D. We prove that the values of h(D) over all labelled acyclic digraphs D on n vertices are asymptotically normally distributed with mean Cn and variance C′n, where C ≈ 0·764334 and C′ ≈ 0·145210. Furthermore, define V0(D) to be the set of sinks (vertices of out-degree 0) and, for r ≥ 1, define Vr(D) to be the set of vertices ν such that the longest directed path from ν to V0(D) has length r. For each k ≥ 1, let nk(D) be the number of sets Vt(D) which have size k. We prove that, for fixed k, the values of nk(D) over all labelled acyclic digraphs D on n vertices are asymptotically normally distributed with mean Ckn and variance C′kn, for positive constants Ck and C′k. Results of Bender and Robinson imply that our claim holds also for unlabelled acyclic digraphs.