A min-max relation in flowgraphs and some applications

A flowgraph G=(V,A,s) is a digraph such that there is a dipath from s to u for every vertex u in G. A vertex w dominates a vertex u if and only if all dipaths in G from s to u pass through w. The vertex s dominates trivially any vertex in G. In this work, we define two sets called dominator cover and junction partition. A dominator cover in G is a set of vertices that includes s and non-trivial dominators for all vertices in G. A junction partition B={B1,…,Bk} is a partition of V which satisfies the following property: if vertex u belongs to Bi and vertex v belongs to Bj (i≠j), then there are internally vertex-disjoint dipaths from s to u and from s to v, for short, s is a junction of u and v. The first part of this work shows that, in flowgraphs, the minimum size of a dominator cover is equal to the maximum size of a junction partition. The second part describes some applications for this relation, such as, a new correctness proof of an algorithm that finds a maximum junction partition in reducible flowgraphs; and good time and space complexity algorithms for problems related to junctions and nearest common ancestors in acyclic digraphs.