Recursive Construction of the Minimal Digraphs

In a digraph D, a module is a vertex subset M such that every vertex outside M does not distinguish the vertices in M. A digraph D with more than two vertices is prime if O, the single-vertex sets, and V(D) are the only modules in D. A prime digraph D is k-minimal if there is some k-element vertex subset U such that no proper induced subdigraph of D containing U is prime. This concept was introduced by A. Cournier and P. Ille in 1998. They characterized the 1-minimal and 2-minimal digraphs. In 2014, M. Alzohairi and Y. Boudabbous described the 3-minimal triangle-free graphs, and in 2015, M. Alzohairi described a class of 4-minimal triangle-free graphs. In this paper, we give a recursive procedure to construct the minimal digraphs. More precisely, given an integer k, with k >= 3, we give a method for constructing the k-minimal digraphs from the (k - 1)-minimal digraphs.