## 4 The g-Extra Conditional Diagnisability of the BCCC Network under the MM\({}^{\star}\) Model
Since \(F_{1}\) and \(F_{2}\) are undistinguishable, they do not comply with Lemma 1(1). Thus, for any vertex \(h\) in \(H\), we have \(N_{F_{1}\triangle F_{2}}(h)=\emptyset\). We infer that \(N_{BC_{n,k}}(H)\subseteq F_{1}\cap F_{2}\). Since \(F_{1}\) is a g-extra fault set, it can be obtained that every component \(J_{i}\in S\cup[F_{2}-(F_{1}\cap F_{2})]\) has \(|V(J_{i})|\geq g+1\). Similarly, \(F_{2}\) is a g-extra fault set, and every component \(K_{i}\in S\cup[F_{1}-(F_{1}\cap F_{2})]\) has \(|V(K_{i})|\geq g+1\). In summary, every component \(G_{i}\in S\cup[F_{2}-(F_{1}\cap F_{2})]\cup[F_{1}-(F_{1}\cap F_{2})]\) has \(|V(G_{i})|\geq g+1\). Since \(F_{1}\) is a g-extra fault set, we have \(|V(H_{i})|\geq g+1\). Therefore, \(F_{1}\cap F_{2}\) is a g-extra vertex cut (see Figure 4). Since \(F_{1}\) and \(F_{2}\) are two distinguishable faulty sets, \(F_{1}\
ot\subseteq F_{2}\) and \(F_{2}\
ot\subseteq F_{1}\), we can induce that \(|F_{1}-F_{2}|\geq 1\) and \(|F_{2}-F_{1}|\geq 1\). Since \(F_{1}\) is a g-extra vertex cut and \((F_{1}\bigtriangleup F_{2})\cap H=\emptyset\), we have \(|(F_{2}-F_{1})\cup S|\geq g+1\). Combining this with \(|F_{1}-F_{2}|\geq 1\), \(|(F_{2}-F_{1})\cup S|+|F_{1}-F_{2}|=|(F_{1}\bigtriangleup F_{2})\cup S|\geq g+2\). Let \(s^{\prime}\) be any node in \(S^{\prime}\) and \(S^{\prime}=(F_{1}\bigtriangleup F_{2})\cup S\). Recall that the vertex set pair \((F_{1},F_{2})\) is not satisfied with any one condition in Lemma 1, \(V(S^{\prime})\cap V(H)=\emptyset\), we have \(N_{BC_{n,k}}(s^{\prime})\subseteq F_{1}\cup F_{2}\). Let \(N_{BC_{n,k}}(S^{\prime})=D\). Thus, \(|F_{1}\cup F_{2}|\geq|D|\). According to Theorem 1, \(s_{g+1}(BC_{n,k})=(g+2)(k-1)+n\geq s_{g}(BC_{n,k})+g\). Thus, \(|D|\geq s_{g}(BC_{n,k})+g\). It can be obtained that \(|F_{2}|=|F_{2}-F_{1}|+|F_{1}\cap F_{2}|\geq s_{g}(BC_{n,k})+g+1\), a contradicition to \(|F_{2}|\leq s_{g}(BC_{n,k})+g\). We can conclude that \(S=\emptyset\). There are no isolated nodes in \(V(BC_{n,k})-F_{1}\cup F_{2}\). \(\alpha(BC_{n,k})=0\). The lemma holds. Theorem 4.2: _For any integers \(n\geq 3\), \(k\geq 2\), \(n\geq k+1\) and \(1\leq g\leq n-1\), the g-extra conditional diagnosability of \(BC_{n,k}\) under the MM\({}^{*}\) model is \(t_{g}(BC_{n,k})=(g+1)k+n\)._  
Proof. : According to Lemma 11, the maximal independent set \(\alpha(BC_{n,k})=0\). Let \(f(g)=N-2x_{g}(BC_{n,k})-2g=(k+1)n^{k+1}-2[(g+1)(k-1)]-2n-2g\). Thus, \(f^{\prime}(g)=-2k<0\) when \(k\geq 2\). So, \(f(g)\) is a decreasing function. \(f(g)\geq f(1)\geq 0\). The graph \(BC_{n,k}\) satisfies with \(N\geq 2x_{g}(BC_{n,k})+2g+\alpha(BC_{n,k})\) and \(\delta(BC_{n,k})\geq 3\) when \(n\geq 3\), \(k\geq 2\), \(n\geq k+1\) and \(1\leq g\leq n-1\). According to Theorem 1, \(\kappa_{g}(BC_{n,k})=N_{BC_{n,k}}(H)\), where \(|V(H)|=g+1\), satisfying condition (1) of Lemma 10. Meanwhile, since \(\kappa_{g}(BC_{n,k})\geq g+2\) and \(\kappa_{g+1}(BC_{n,k})\geq\kappa_{g}(BC_{n,k})+g\), this satisfies condition (2) of Lemma 10. According to Lemma 7, \(mc(BC_{n,k}-F)\geq N-|F|-(g+1)\), where \(|F|\leq\kappa_{g}(BC_{n,k})-1\leq\kappa_{g+1}(BC_{n,k})-1\), satisfying condition (3) of Lemma 10. Therefore, \(t_{g}(BC_{n,k})=\kappa_{g}(BC_{n,k})+g=(g+1)k+n\). The theorem holds.