  Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is
the polynomial $D(G, x)=\sum_{i=1}^n d(G,i) x^i$, where $d(G,i)$ is the number
of dominating sets of $G$ of size $i$. The $n$-barbell graph $Bar_n$ with $2n$
vertices, is formed by joining two copies of a complete graph $K_n$ by a single
edge. We prove that for every $n\geq 2$, $Bar_n$ is not $\mathcal{D}$-unique,
that is, there is another non-isomorphic graph with the same domination
polynomial. More precisely, we show that for every $n$, the
$\mathcal{D}$-equivalence class of barbell graph, $[Bar_n]$, contains many
graphs, which one of them is the complement of book graph of order $n-1$,
$B_{n-1}^c$. Also we present many families of graphs in
$\mathcal{D}$-equivalence class of $K_{n_1}\cup K_{n_2}\cup \cdots\cup
K_{n_k}$.
