Counting directed acyclic and elementary digraphs
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Directed acyclic graphs (DAGs) can be characterised as directed graphs whose strongly connected components are isolated vertices. Using this restriction on the strong components, we discover that when m = cn, where m is the number of directed edges, n is the number of vertices, and c < 1, the asymptotic probability that a random digraph is acyclic is an explicit function p(c), such that p(0) = 1 and p(1) = 0. When m = n(1 + μ n^-1/3), the asymptotic behaviour changes, and the probability that a digraph is acyclic becomes n^-1/3 C(μ), where C(μ) is an explicit function of μ. Łuczak and Seierstad (2009, Random Structures Algorithms, 35(3), 271–293) showed that, as μ→ -∞, the strongly connected components of a random digraph with n vertices and m = n(1 + μ n^-1/3) directed edges are, with high probability, only isolated vertices and cycles. We call such digraphs elementary digraphs. We express the probability that a random digraph is elementary as a function of μ. Those results are obtained using techniques from analytic combinatorics, developed in particular to study random graphs.
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