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Let $A,B,Z$ be pairwise disjoint sets of vertices in a directed acyclic graph. A graph path, traversed without requiring its edges to point in the direction of travel, is active given $Z$ precisely when every interior noncollider lies outside $Z$ and every collider either belongs to $Z$ or has a descendant in a directed graph in $Z$. The sets $A,B$ are D-separated by $Z$ when no active graph path connects them.^[ref]

In the three-vertex chain $X\to M\to Y$ and fork $X\leftarrow M\to Y$, conditioning on $M$ blocks the only graph path. In $X\to M\leftarrow Y$, the collider blocks the graph path before conditioning; conditioning on $M$ or its descendant in a directed graph activates it. Other vertices can still block a longer graph path.

Under the global Markov property of a directed acyclic graph, D-separation implies conditional independence. The reverse implication for a particular probability distribution requires faithfulness of a directed acyclic graph: additional conditional independences can arise through special parameter choices.