The Duo-Free and Critically Duo-Free Graphs

Abstract

Let G(V, A) be a digraph. A subset M ⊂ V is a module of G if each point outside M has the same behavior to all points of M. Clearly, the empty set, the singleton and the set of vertices V are modules of G known as trivial modules. A digraph is indecomposable if all its modules are trivial, otherwise it is decomposable. The central notion of this work is the duo which is a module of cardinality 2. The digraphs which do not admit any duo are called duo-free digraphs. An important tool in this work is the notion of duo-free-minimal digraphs defined as follows : Let G be a duo-free digraph. G is duo-free-minimal for elements x1, ..., xk of V if for each proper subset X of V containing x1, ..., xk , the induced subdigraph G[X] has at least one duo. The objective of our study is to present the characterization of the duo-free-minimal tournaments and graphs for one or two vertices. Moreover, we describe the duo-free minimal k-vertex digraph for one vertex where k ∈ {3, 4, 5}.