Lecture notes on “Analysis of Algorithms”: Directed Minimum Spanning Trees

We say that u is dominated by v if and only if every path from r to u passes through v. It is not difficult to show that if u is dominated by both v1 and v2, then either v1 dominates v2, or v2 dominates v1. We say that v is the immediate dominator of u if and only if every dominator of u is also a dominator of v. It follows from the previous observation that every vertex u has a unique immediate dominator, which might be r. We define the domination tree of the graph as a tree in which the parent of each vertex other than r is its immediate dominator. There are linear-time algorithms for constructing the domination tree of a graph with respect to a given root.