Abstract As a fundamental metric, the connectivity to assess fault tolerance and reliability of interconnection networks has been extensively explored. However, classical connectivity is not very effective at evaluating large-scale networking systems. To overcome this deficiency, two new indices, cluster connectivity and super cluster connectivity, have been proposed to characterize the robustness of interconnection networks. This paper focuses on investigating $\mathcal{H} (\mathcal{H}^{*})$-cluster connectivity and super $\mathcal{H} (\mathcal{H}^{*})$-connectivity of composition graph $DQ_{n}$, based on disc-ring and hypercube, for $\mathcal{H}\in \{K_{1,r}\ |\ 0\leq r\leq n+1\}$, respectively. In detail, we show that $\kappa (DQ_{n}|K_{1,1} (K_{1,1}^{*}))=\kappa ^{\prime}(DQ_{n}|K_{1,1}(K_{1,1}^{*})) =n+1 (n\geq 3)$, $\kappa (DQ_{n}|K_{1,r}(K_{1,r}^{*})) =\left\lceil \frac{n}{2}\right\rceil +1 (2\leq r\leq 4)$ for $n\geq 3$, $\kappa ^{\prime}(DQ_{n}|K_{1}(K_{1}^{*})) =\kappa ^{\prime}(DQ_{n})=2n (n\geq 3)$, and for $2\leq r\leq 3$ and $k\geq 2$, $$\begin{align*} \kappa^{\prime}(DQ_{n}|K_{1,r}(K_{1,r}^{*}))=\left\{\begin{array}{@{}ll} n+1, & if\ n=2k+1; \\ n, & if\ n=2k. \end{array}\right. \end{align*}$$As by-products, we show that DQcube is super $K_{1,r}-\ (K_{1,r}^{*}-)$connected $(2\leq r\leq 3)$, and derive the 4-extra connectivity of DQcube $\kappa _{4}(DQ_{n}) = 5n-9 (n\geq 4)$.