## 4 Restricted Connectivity of BCCC
Let \(x\) be the neighbor of \(u_{2}\) and \(N_{G}(x)\cap V(BCC^{R}_{n,k})\
eq\emptyset\). Let \(y_{1},y_{2},\ldots,y_{n-1}\) be the same type \(A\) switch neighbors of \(u_{2};y_{1},y_{2},\ldots,y_{n-1^{\prime}}\) be all neighbors of \(y_{1},y_{2},\ldots,y_{n-1}\) in \(BCC^{a}_{n,k}\), and \(N_{G}(y_{i^{\prime}})\cap V(BCC^{R}_{n,k})\
eq\emptyset,i\in\{1,2,\ldots,n-1\}\). Let \(z_{1},z_{2},\ldots,z_{k-1}\) be neighbors of \(u_{2}\) in the same type \(B\) switch, except \(x\); \(z_{1^{\prime}},z_{2^{\prime}},\ldots,z_{k-1^{\prime}}\) be the same type \(A\) switch neighbors of \(z_{1},z_{2},\ldots,z_{k-1}\); \(z_{1^{\prime\prime}},z_{2^{\prime\prime}},\ldots,z_{k-1^{\prime\prime}}\) be all neighbors of \(z_{1^{\prime}},z_{2^{\prime}},\ldots,z_{k-1^{\prime}}\) in \(BCC^{a}_{n,k}\) and \(N_{G}(z_{i^{\prime\prime}})\cap V(BCC^{R}_{n,k})\
eq\emptyset,i\in\{1,2,\ldots, k-1\}\). Above all, there are \((k-1)(n-1)+n\) paths from \(u_{2}\) to \(BCC^{R}_{n,k}\) as follows (refer to Figure 4). The specific paths are listed below:  
\[ \begin{cases}X_{i}=(u_{2},x)\\ Y_{i}=(u_{2},y_{i},y_{i^{\prime}}),1\leq i\leq n-1\\ Z_{i}=(u_{2},z_{i},z_{i^{\prime}},z_{i^{\prime\prime}}),1\leq i\leq k-1,i^{ \prime}=(k-1)(n-1)\end{cases} \]  
Since \(|F(G)|=2k(G)-n-1=2(n+k-1)-n-1=n+2k-3\), we have \((k-1)\)\((n-1)+n-|F(G)|=1\) when \(n=k+1\) and \(n=3\). Furthermore, when \(n>3\), whether \(\kappa^{1}(G)=2\kappa(G)-n\) or \(2\kappa(G)-k-1\), \((k-1)(n-1)+n-\kappa^{1}(G)>1\). Therefore, there is at least one fault-free path \(P\) in the above \((k-1)(n-1)+n\) paths from \(u_{2}\) to \(BCC^{R}_{n,k}\). Assume that \(w\) is the last vertex in the path \(P_{1}\). If \(w=v\), then \(P_{1}\) is a path from \(u\) to \(v\) in \(G-F\). Otherwise, since \(BCC^{R}_{n,k}-F_{R}\) is connected, there is path \(P_{2}\) between \(w\) and \(v\) in \(BCC^{R}_{n,k}\). Then, the path \((P_{1},P_{2})\) is a path from \(u_{2}\) to \(v\) in \(G-F\). Subcase 1.2. \(|F_{a}|<\kappa(BCC^{a}_{n,k})\) and \(|F_{\beta}|<\kappa(BCC^{\beta}_{n,k})\). Because \(BCC^{a}_{n,k}-F_{a}\) and \(BCC^{\beta}_{n,k}-F_{\beta}\) are all connected, let \(\{x_{i}|x_{1},x_{2},\ldots,x_{n^{2}}\}\) be a set of vertices connected to other subgraphs in \(BCC^{a}_{n,k}\). Let \(\{y_{i}|y_{1},y_{2},\ldots,y_{n^{2}}\}\) be a set of vertices connected to other subgraphs in \(BCC^{\beta}_{n,k}\). It can be obtained that \(|N_{G}(x_{i})\cap y_{i}|=n^{2}\). Whether \(\kappa^{1}(G)=2\kappa(G)-n\) or \(2\kappa(G)-k-1,n^{2}>\kappa^{1}(G)-1\). Therefore, there must exist a fault-free vertex \(x(x\in x_{i})\) and \(y(y\in y_{i})\) in the same type A switch. If \(u=x\), then it is directly connected to \(BCC^{\beta}_{n,k}\). If \(u\
eq x\), since \(BCC^{a}_{n,k}-F_{a}\) is connected, there exists a fault-free path \(P_{1}\) from \(u\) to \(x\) in \(BCC^{a}_{n,k}-F_{a}\). Since \(x\) and \(y\) are located in the same switch, there is a fault-free path \(P_{2}\) from \(x\) to \(y\). If \(v=y\), it is directly connected to \(BCC^{a}_{n,k}\). If \(v\
eq y\), since \(BCC^{\beta}_{n,k}-F_{\beta}\) is connected, there exists a fault-free path \(p_{3}\) from \(y\) to \(v\) in \(BCC^{\beta}_{n,k}-F_{\beta}\). Then, the path \((P_{1},P_{2},P_{3})\) is a path from \(u\) to \(v\) in \(G-F\) (refer to Figure 5b). Subcase 2. \(\alpha=\beta\). If \(|F_{a}|<\kappa(BCC^{a}_{n,k})\), \(BCC^{a}_{n,k}-F_{a}\) is connected. There exists a fault-free path from \(u\) to \(v\) in \(BCC^{a}_{n,k}\). Thus, we only need to consider that \(|F_{a}|\geq\kappa(BCC^{a}_{n,k})\). According to subcase 1.1, if \(BCC^{a}_{n,k}-F_{a}\) is not connected, \(BCC^{R}_{n,k}-F_{R}\) is connected. According to subcase 1.1.1 and subcase 1.1.2, whether \(u=u_{1}\) or \(u=u_{2}\), there must exist a fault-free path from \(u(v)\) to \(BCC^{R}_{n,k}\). Thus, there exists a fault-free path \(P_{1}\) from \(u\) to a vertex \(x\) in \(BCC^{a}_{n,k}\) and a fault-free path \(P_{2}\) from \(v\) to a vertex \(y\) in \(BCC^{a}_{n,k}\). If \(x=y\), the path \((P_{1},P_{2}^{-1})\) is a path from \(u\) to \(v\).