Characterizations of Reducible Flow Graphs

  • Authors:

  • M. S. Hecht;J. D. Ullman

  • Affiliations:

  • Department of Computer Science, University of Maryland, College Park, Maryland;Department of Electrical Engineering, Princeton University, Princeton, New Jersey

  • Venue:
  • Journal of the ACM (JACM)
  • Year:
  • 1974

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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, n0) 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 E1 and E2 such that E1 forms a dag D of G and each (n, m) in E2 has n = m or m dominates n in G. (d) Same as (c), except each (n, m) in E2 has n = m or m dominates n in D. (e) Same as (c), except E2 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.