Dominators, Directed Bipolar Orders, and Independent Spanning Trees

We consider problems related to dominators and independent spanning trees in flowgraphs and provide linear-time algorithms for their solutions. We introduce the notion of a directed bipolar order, generalizing a previous notion of Plein and Cheriyan and Reif. We show how to construct such an order from information computed by several known algorithms for finding dominators. We show how to concurrently verify the correctness of a dominator tree D and a directed bipolar order O very simply, and how to construct from D and O two spanning trees whose paths are disjoint except for common dominators. Finally, we describe alternative ways to verify dominators without using a directed bipolar order.