Character degree graphs with no complete vertices

Let Γ be a graph in which each vertex is non-adjacent to another different one. We show that, if G is a finite solvable group with abelian Fitting subgroup and with character degree graph Γ(G)=Γ, then G is a direct product of subgroups having a disconnected character degree graph. In particular, Γ is a join of disconnected graphs. We deduce also that solvable groups with abelian Fitting subgroup have a character degree graph with diameter at most 2.