Character degree graphs and normal subgroups

We consider the degrees of those irreducible characters of a group $G$whose kernels do not contain a given normal subgroup $N$. We show that if $N \subseteq G'$ and $N$ is not perfect, then the common-divisor graph on this set of integers has at most two connected components. Also, if $N$ is solvable, we obtain bounds on the diameters of the components of this graph and, in the disconnected case, we study the structure of $N$ and of $G$.