A characterization of Markov equivalence for directed cyclic graphs

The concept of d-separation (Pearl, 1988) was originally defined for acyclic directed graphs, but there is a natural extension of the concept to directed graphs with cycles. When exactly the same set of d-separation relations hold in two directed graphs, no matter whether respectively cyclic or acyclic, we say that they are Markov equivalent In other words, when two directed cyclic graphs are Markov equivalent the set of distributions that satisfy a natural extension of the global directed Markov condition (Lauritzen et al, 1990) is exactly the same for each graph. There is an obvious exponential (in the number of vertices) time algorithm for deciding Markov equivalence of two directed cyclic graphs: simply check all of the d-separation relations in each graph. In this paper I prove a theorem that gives necessary and sufficient conditions for two directed cyclic graphs to be Markov equivalent where each of the conditions can be checked in polynomial time. Hence, the theorem can be easily adapted into a polynomial time algorithm for deciding the Markov equivalence of two directed cyclic graphs (Richardson, 1996). (C) 1997 Elsevier Science Inc.