 (i) The Minimality Condition. Suppose that the acyclic directed graph \(\bG\) on variable set \(\bV\) satisfies MC with respect to the probability distribution P. The Minimality Condition asserts that no sub-graph of \(\bG\) over \(\bV\) also satisfies the Markov Condition with respect to P. (A subgraph of \(\bG\) is a graph over \(\bV\) that results from removing arrows from \(\bG)\). As an illustration, consider the variable set \(\{X, Y\}\), let there be an arrow from X to Y, and suppose that X and Y are probabilistically independent of each other according to probability function P. This graph would satisfy the MC with respect to P: none of the independence relations mandated by the MC are absent (in fact, the MC mandates no independence relations). But this graph would violate the Minimality Condition with respect to P, since the subgraph that omits the arrow from X to Y would also satisfy the MC. The Minimality Condition implies that if there is an arrow from X to Y, then X makes a probabilistic difference for Y, conditional on the other parents of Y. In other words, if \(\bZ = \PA(Y) \setminus \{X\}\), there exist \(\bz\), y, x, \(x'\) such that