Domination polynomial of lexicographic product of specific graphs

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G,\lambda )={\sum }_{i=0}^{n}d(G,i){\lambda }^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. We consider the lexicographic product of two specific graphs and study their domination polynomials.