  Chain graphs (CG) use undirected and directed edges to represent both
structural and associative dependences. Like acyclic directed graphs (ADGs),
the CG associated with a statistical Markov model may not be unique, so CGs
fall into Markov equivalence classes, which may be superexponentially large,
leading to unidentifiability and computational inefficiency in model search and
selection.Under the Andersson--Madigan--Perlman (AMP) interpretation of a CG,
each Markov-equivalence class can be uniquely represented by a single CG, the
AMP essential graph, that can be used for efficient inference and model search
and selection. Here the graphical structure of an AMP essential graph is
studied in detail. This structural information will be useful for algorithmic
construction of the essential graph.
