Fault diagnosis is essential for maintaining the regular operation of multiprocessor systems. In this paper, we introduce a novel concept of$r$-component$H$-structure diagnosability, denoted as$t_{s}^{r}(G;H)$, which represents the maximum number of$r$-component$H$-structure faults that system$G$can precisely detect. This parameter enables us to assess the overall diagnosability of the system. Furthermore, we strictly prove that under the PMC and MM* diagnostic models, the following results hold: for$n\geq 7$,$t_{s}^{2}(Q_{n};K_{1,1})=2n-4$; for$n\geq 6$,$t_{s}^{2}(Q_{n};K_{1,2})=n-2$. These results confirm that when the system experiences a$K_{1,1}$-structural fault and is disconnected at this time, the maximum number of detectable$K_{1,1}$-structures within the system is guaranteed to be twice the original number in previous studies.