It is a key step of the junction tree algorithm, used in belief propagation on graphical models.
The moralized counterpart of a directed acyclic graph is formed by adding edges between all pairs of non-adjacent nodes that have a common child, and then making all edges in the graph undirected.
The name stems from the fact that, in a moral graph, two nodes that have a common child are required to be married by sharing an edge.
A chordal graph (a.k.a., recursive simplicial) is a special case of weakly recursively simplicial when no edge is removed during the elimination process.
[2] Unlike chordal graphs that can be recognised in polynomial time, Verma & Pearl (1993) proved that deciding whether or not a graph is moral is NP-complete.