Splitting a graph by a given partition of the set of vertices based on the minimum weight of the induced trees

A method for considering a weighted directed graph with an accuracy of up to a given partition of the set of vertices is proposed. The resulting digraph (the splitting graph) does not contain arcs inside each partition element, and the arcs between the partition atoms are calculated in a special way taking into account the arcs of the original directed graph inside the atoms. This accounting is based on minimal trees defined on atoms. A study was made of what information about the original digraph is preserved in its splitting.