  A concentration graph associated with a random vector is an undirected graph
where each vertex corresponds to one random variable in the vector. The absence
of an edge between any pair of vertices (or variables) is equivalent to full
conditional independence between these two variables given all the other
variables. In the multivariate Gaussian case, the absence of an edge
corresponds to a zero coefficient in the precision matrix, which is the inverse
of the covariance matrix. It is well known that this concentration graph
represents some of the conditional independencies in the distribution of the
associated random vector. These conditional independencies correspond to the
"separations" or absence of edges in that graph. In this paper we assume that
there are no other independencies present in the probability distribution than
those represented by the graph. This property is called the perfect
Markovianity of the probability distribution with respect to the associated
concentration graph. We prove in this paper that this particular concentration
graph, the one associated with a perfect Markov distribution, can be determined
by only conditioning on a limited number of variables. We demonstrate that this
number is equal to the maximum size of the minimal separators in the
concentration graph.
