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.

中文翻译：

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关于图的独立支配多项式

简单图$G=(V,E)$的独立支配集是$G$中既支配又独立的顶点子集。图 $G$ 的独立支配多项式是多项式 $D_i(G,x)=\sum_{A} x^{|A|}$，对所有独立支配子集 $A\subseteq V$ 求和。$D_i(G,x)$ 的根称为独立支配根。我们研究了一些广义复合图的独立支配多项式。结果，我们构建了独立支配根为实数的图。此外，我们考虑一些特定的图并研究它们独立支配集的数量。