Running intersection property ($C_i\cap C_j\subseteq C_k\text{along the path from }i\text{ to }j$)

The running intersection property says that every variable belongs to a connected subtree of a junction tree. Equivalently, every clique along the path between two cliques contains their intersection. This ensures that eliminating variables on one side of a tree edge leaves a message depending only on its separator with the other side.