Algorithm for transforming an oriented graph into an acyclic one

Over the past five years together with experts in various fields of knowledge we have solved a number of urgent problems. Using elements of graph theory, we constructed original economic algorithms to study the models under consideration and tested them on real data. The article is a continuation of researches in this field, and the current task has been formulated by experts in bioengineering. At the first stage, a factorization by equivalence relation was carried out in the digraph, we identified input and output vertices (of inward and outward edges) in each cluster using the previously developed by us economic algorithm. At the second stage, each cluster was replaced by means of an algorithm of wave front with its acyclic subgraph connecting the input vertices with the output paths of the minimal length. Further, the initial digraph was replaced by an acyclic digraph without feedbacks connecting input and output vertices of clusters.