Characterizations of Reducible Flow Graphs

Abstract

It is established that if G is a reducible flow graph, then edge ( n, m ) is backward (a back latch) if and only if either n = m or m dominates n in G . Thus, the backward edges of a reducible flow graph are unique. Further characterizations of reducibility are presented. In particular, the following are equivalent: (a) G = ( N, E, n 0 ) is reducible. (b) The “dag” of G is unique. (A dag of a flow graph G is a maximal acyclic flow graph which is a subgraph of G .) (c) E can be partitioned into two sets E 1 and E 2 such that E 1 forms a dag D of G and each ( n, m ) in E 2 has n = m or m dominates n in G . (d) Same as (c), except each ( n, m ) in E 2 has n = m or m dominates n in D . (e) Same as (c), except E 2 is the back edge set of a depth-first spanning tree for G . (f) Every cycle of G has a node which dominates the other nodes of the cycle. Finally, it is shown that there is a “natural” single-entry loop associated with each backward edge of a reducible flow graph.