  Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of
$G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some
vertices of $D$. The total domination number of $G$ is equal to minimum
cardinality of total dominating set in $G$ and is denoted by $\gamma_t(G)$. The
total domination polynomial of $G$ is the polynomial
$D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)x^i$, where $d_t(G,i)$ is the number
of total dominating sets of $G$ of size $i$. Two graphs $G$ and $H$ are said to
be total dominating equivalent or simply $\mathcal{D}_t$-equivalent, if
$D_t(G,x)=D_t(H,x)$. The equivalence class of $G$, denoted $[G]$, is the set of
all graphs $\mathcal{D}_t$-equivalent to $G$. In this paper, we investigate
$\mathcal{D}_t$-equivalence classes of some graphs. Also we introduce some
families of graphs whose total domination polynomials are unimodal.
