A contraction algorithm for finding small cycle cutsets

We consider the problem of finding a minimum size cutset in a directed graph G = (V, E), i.e., a vertex set that cuts all cycles in G. Since the general problem is NP-complete we concentrate on finding small cutsets. The algorithm we suggest uses contraction operations to reduce the graph size and to identify candidates for the cutset; the complexity of the algorithm is O(|E|log|V|). This contraction algorithm is compared to Shamir-Rosen algorithm. It is shown that the class of graphs for which the contraction algorithm finds a minimum cutset (completely contractible graphs) properly contains the class of graphs for which Shamir-Rosen algorithm finds a minimum cutset (quasi-reducible graphs) and thus that the contraction algorithm is more powerful. As a by-product of this analysis we construct a hierarchy of the classes of graphs for which minimum cutsets can be found efficiently. The class of quasi-reducible graphs lies, in this hierarchy, between two classes which are closely related. This result illuminates the nature of the quasi-reducible graphs. The hierarchy constructed allows us also to compare the Wang-Lloyd-Soffa algorithm to the Shamir-Rosen algorithm and to the contraction algorithm.