Efficient Construction of B -essential Graphs

Essential graphs and largest chain graphs are well-established graphical representations of equivalence classes of DAGs and chain graphs (CGs) respectively, especially useful in the context of structural learning. Recently, Roverato and La Rocca (2004) introduced the notion of a labelled block ordering of vertices B as a flexible tool for specifying subfamilies of CGs. In particular, both the family of DAGs and the family of “unconstrained” CGs are classes of B -consistent CGs for the appropriate choice of B . Equivalence classes of B -consistent CGs are represented by means of B -essential graphs. In this paper, we give an efficient procedure for the construction of B -essential graphs that is based on the idea of a topological sorting of meta-arrows. In this way we also provide an efficient procedure for the construction of both largest chain graphs and essential graphs. The key feature of the proposed procedure is that every meta-arrow needs to be processed only once.