The union of all sets equals . That is, each graph vertex is associated with at least one tree node. For every edge in the graph, there is a subset that contains both and . That is, vertices are adjacent in the graph only when the corresponding subtrees have a node in common. If and both contain a vertex , then all nodes of the tree in the (unique) path between and contain as well. That is, the nodes associated with vertex form a connected subset of . This is also known as coherence, or the running intersection property. It can be stated equivalently that if , and are nodes, and is on the path from to , then .
The union of all sets equals . That is, each graph vertex is associated with at least one tree node For every edge in the graph, there is a subset that contains both and . That is, vertices are adjacent in the graph only when the corresponding subtrees have a nod.e in common. If and both con.tain a vertex , then all nodes of the tree in the (unique) path between and contain as well. That is, the nodes associated with vertex form a connected subset of . This is also known as coherence, or the running intersection property. It can be stated equivalently that if , and are nodes, and is on the path from to , then 