The matter of concern are algorithms for the discrimination of direct from indirect regulatory effects from an interaction graph built up by error-prone measurements. Many of these algorithms can be cast as a rule for the removal of a single edge of the graph, such that the remaining graph is still consistent with the data. A set of mild conditions is given under which iterated application of such a rule leads to a unique minimal consistent graph. We show that three of the common methods for direct interactions search fulfill these conditions, thus providing a justification of their use. The main issues a reconstruction algorithm has to deal with, are the noise in the data, the presence of regulatory cycles, and the direction of the regulatory effects. We introduce a novel rule that, in contrast to the previously mentioned methods, simultaneously takes into account all these aspects. An efficient algorithm for the computation of the minimal graph is given, whose time complexity is cubic in the number of vertices of the graph. Finally, we demonstrate the utility of our method in a simulation study.