  Moral graphs were introduced in the 1980s as an intermediate step when
transforming a Bayesian network to a junction tree, on which exact belief
propagation can be efficiently done. The moral graph of a Bayesian network can
be trivially obtained by connecting non-adjacent parents for each node in the
Bayesian network and dropping the direction of each edge. Perhaps because the
moralization process looks simple, there has been little attention on the
properties of moral graphs and their impact in belief propagation on Bayesian
networks. This paper addresses the mistaken claim that it has been previously
proved that optimal moral graph triangulation with the constraints of minimum
fill-in, treewidth or total states is NP-complete. The problems are in fact
NP-complete, but they have not previously been proved. We now prove these.
