## 7 Conclusions  
The diagnosability is the maximum number of fault processors that the system can guarantee to be diagnosed irreplaceably. It plays an important role in measuring the reliability and fault tolerance of the network. Meanwhile, a good fault diagnosis algorithm can effectively diagnose and exclude the fault nodes in the network. In this paper, we first prove that the g-extra connectivity of \(BC_{n,k}\) is \(\kappa_{g}(BC_{n,k})=(g+1)(k-1)+n\), where \(0\leq g\leq n-1\). Next, we establish the g-extra connectivity of \(BC_{n,k}\) under the MM* model. We obtain that \(t_{g}(BC_{n,k})=(g+1)k+n\), where \(1\leq g\leq n-1\). Based on the results of extra connectivity and the properties of the largest connectivity component, we study the t/k-diagnosability of BCCC network equal to the g-extra connectivity under the MM* model where \(1\leq k\leq n-1\). Furthermore, we give a t/k diagnosis algorithm, which can correctly identify all nodes except most k nodes undiagnosed under the MM* model. The time complexity of the algorithm is \((N(n+k-1))\). Finally, we have proven the accuracy and reliability of the algorithm through experiments. So far, t/k diagnosis algorithms for most interconnection networks under the MM* model have not been studied. We hope to give you some enlightenment with our proof. What is more, the restricted edge connectivity and g-good-neighbor local diagnosability were studied in [28; 29]. These new diagnostic strategies can divide large networks into smaller networks and analyze the diagnostic degree one by one. By proposing new diagnostic algorithms based on this strategy, the entire network can be diagnosed with more faulty nodes based on the new algorithms. The local diagnosability of BCCC networks can be investigated in the future. Conceptualization, [PERSON] ([PERSON]); Methodology, [PERSON] ([PERSON]); Software, W.Z. ; Formal analysis, W.Z. and J.L. ([PERSON]); Investigation, W.Z. and J.L. ([PERSON]); Resources, [PERSON] ([PERSON]); Data curation, [PERSON] ([PERSON]); Writing -- original draft, [PERSON] ([PERSON]); Writing--review and editing, [PERSON] ([PERSON]) and J.L. ([PERSON]); Project administration, W.Z. and J.L. ([PERSON]); Funding acquisition, [PERSON] ([PERSON]). All authors have read and agreed to the published version of the manuscript. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \(BC_{3,2}\) & \(BC_{4,2}\) & \(BC_{4,3}\) & \(BC_{5,3}\) & \(BC_{5,4}\) \\ \hline \multirow{2}{*}{h = 1} & Number of fault nodes & 5 & 6 & 8 & 9 & 11 \\ \cline{2-6}  & Number of fault nodes diagnosed & 5 & 6 & 8 & 9 & 11 \\ \hline \multirow{2}{*}{h = 2} & Number of fault nodes & 6 & 7 & 10 & 11 & 14 \\ \cline{2-6}  & Number of fault nodes diagnosed & 6 & 7 & 10 & 11 & 14 \\ \hline \multirow{2}{*}{h = 3} & Number of fault nodes & 7 & 8 & 12 & 13 & 17 \\ \cline{2-6}  & Number of fault nodes diagnosed & 7 & 8 & 12 & 13 & 17 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of fault nodes executing algorithm t/h-diag diagnosis. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Randomly Generated Fault Nodes** & **Fault Node Diagnosed** \\ \hline \multirow{2}{*}{\(BC_{3,2}\)} & h = 1 & 0202, 1112, 2021, 1101, 2211 & 0202, 1101, 2021, 1112, 2211 \\ \cline{2-5}  & h = 2 & 0211, 2011, 0121, 2120, 0212, 0120 & 0211, 0121, 0120, 2011, 0212, 2120 \\ \hline \multirow{2}{*}{\(BC_{4,2}\)} & h = 1 & 0210, 0331, 1032, 0312, 1110, 2032 & 0331, 1032, 2032, 0210, 0312, 1110 \\ \cline{2-5}  & h = 2 & 0012, 1230, 3312, 2231, 2331, 0131, 3202 & 0012, 0131, 3202, 3312, 2231, 2130 \\ \cline{2-5}  & h = 3 & 1321, 3200, 0210, 0130, 2320, 1211, 1000, 0100 & 0100, 1000, 0210, 0130, 3200, 1211, 1321, 2320 \\ \hline \multirow{2}{*}{\(BC_{4,3}\)} & h = 1 & 03132, 01322, 31033, 10200, 13003, 13220, 32322, 30210 & 13003, 10200, 01322, 03132, 31033, 30210, 32322, 13220 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The experimental results obtained by executing the t/h-diag algorithm.