  An independent dominating set of the simple graph $G=(V,E)$ is a vertex
subset that is both dominating and independent in $G$. The independent
domination polynomial of a graph $G$ is the polynomial $D_i(G,x)=\sum_{A}
x^{|A|}$, summed over all independent dominating subsets $A\subseteq V$. A root
of $D_i(G,x)$ is called an independence domination root. We investigate the
independent domination polynomials of some generalized compound graphs. As
consequences, we construct graphs whose independence domination roots are real.
Also, we consider some certain graphs and study the number of their independent
dominating sets.
