PROPERTIES OF MINIMAL PRIMITIVE DIGRAPHS

It is proved that, for n >= 4, the complexity of the determination of all n-vertex minimal primitive digraphs, which are parts of a given n-vertex primitive digraph Gamma, coincides with the complexity of the recognition of a monotone Boolean function in s variables where s is the number of arcs (i, j) in Gamma such that the vertex i out-degree and the vertex j in-degree exceed 1. It is found that, for n >= 4, all the primitive n-vertex digraphs with n + 1 arcs are minimal graphs and there are minimal primitive n-vertex digraphs with the number of arcs from n + 2 to 2n - 3. Minimal primitive n-vertex digraphs with n + 1 and n + 2 arcs are described.