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Finding the depth of a flow graph

The depth of a flow graph is the maximum number of back edges in a cycle free path, where a back edge is defined by some depth-first spanning tree for the flow graph. In the case of a reducible graph, the depth is independent of the DFST chosen. We show that the computation of the depth of a reducible flow graph may be done in polynomial time. Our algorithm is 0(ne) on a flow graph of n nodes and e edges. Since e≤2n for normal flow graphs, our algorithm is really 0(n2). While even an 0(n2) algorithm is not likely to be acceptable, it is suggestive of the possibility of a more efficient algorithm. Finally, we show that the general problem of computing the depth of an arbitrary flow graph with respect to an arbitrary DFST is NP-complete.