M. Studeny, A. Roverato, S. Stepanova:
Two operations of merging and splitting components in a chain graph.
Kybernetika 45 (2009),
n. 2, pp. 208248.
 Abstract

In this paper we study two operations of merging components in a
chain graph, which appear to be elementary operations yielding an
equivalent graph in the respective sense. At first, we recall
basic results on the operation of feasible merging
components, which is related to classic LWF
(Lauritzen, Wermuth and Frydenberg) Markov equivalence of chain graphs.
These results are used to get a graphical
characterisation of factorisation equivalence of classic
chain graphs. As another example of the use of this operation, we
derive some important invariants of LWF Markov equivalence of
chain graphs. Last, we recall analogous basic results on the
operation of legal merging components. This operation is
related to the socalled strong equivalence of chain
graphs, which includes both classic LWF equivalence and
alternative AMP (Andersson, Madigan and Perlman) Markov equivalence.
 AMS classification 62H05, 68T30, 05C90
 Keywords
 chain graph
 factorisation equivalence
 essential graph
 feasible merging components
 legal merging components
 strong equivalence

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The paper develops the ideas from these papers: