On the structure of random unlabelled acyclic graphs

Abstract

One can use Poisson approximation techniques to get results about the asymptotics of graphical properties on random unlabelled acyclic graphs i.e., on random unlabelled free (rootless) trees. We will use some “colored” partitions to get some rough descriptions of the structure of “most” unlabelled acyclic graphs. In particular, we will prove that for any fixed rooted tree T, almost every sufficiently large acyclic graph has a “subtree” isomorphic to T. We can use this result to get a zero-one law for Monadic Second Order queries on random unlabelled acyclic graphs.