Some Structural Resuits on Prime Graphs

Given a graph G = (V,E), a subset M of V is a module [17] (or an interval [10] or an autonomous [11] or a clan [8] or a homogeneous set [7] ) of G provided that x ∼ M for each vertex x outside M.  So V,φ and {x}, where x ∈ V , are modules of G, called trivial modules. The graph G is indecomposable [16] if all the modules of G are trivial. Otherwise we say that G is decomposable . The prime graph G is an indecomposable graph with at least four vertices. Let G and H be two graphs. Let If G has no induced subgraph isomorphic to H, then we say that G is H-free. In this paper, we will prove the next theorem