  In Pe\~na (2007), MCMC sampling is applied to approximately calculate the
ratio of essential graphs (EGs) to directed acyclic graphs (DAGs) for up to 20
nodes. In the present paper, we extend that work from 20 to 31 nodes. We also
extend that work by computing the approximate ratio of connected EGs to
connected DAGs, of connected EGs to EGs, and of connected DAGs to DAGs.
Furthermore, we prove that the latter ratio is asymptotically 1. We also
discuss the implications of these results for learning DAGs from data.
