## 4 The g-Extra Conditional Diagnisability of the BCCC Network under the MM\({}^{\star}\) Model  
In this section, we will determine the g-extra conditional diagnosability \(t_{g}(BC_{n,k})\) of \(BC_{n,k}\) under the MM\({}^{\star}\) model for \(1\leq g\leq n-1\). Definition 2 ([14]): A simple undirected graph G=(V,E) is \(g\)-extra conditionally t-diagnosable if and only if for each pair of distinct faulty g-extra vertex sets \(F_{1}\), \(F_{2}\subseteq V(G)\) such that \(|F_{1}|\leq t\), \(|F_{2}|\leq t\), \(F_{1}\) and \(F_{2}\) are distinguishable. The \(g\)-extra conditional diagnosability of \(G\), denoted as \(t_{g}(G)\), is the maximum value of \(t\) such that \(G\) is \(g\)-extra conditionally t-diagnosable. Lemma 10 ([26]): _Let \(G\) be a connected graph with order \(N\geq 2\kappa_{g}(G)+2g+\alpha(G)\) and minimum degree \(\delta(G)\geq 3\), where \(\kappa_{g}(G)\) is the \(g\)-extra connectivity of \(G\), \(\alpha(G)\) is the independence number of \(G\) and \(g\geq 0\). If \(G\) satisfies the following conditions:_  
1. _there is a connected subgraph_ \(H\) _of_ \(G\) _with_ \(|V(H)|=g+1\) _such that_ \(N_{G}(H)\) _is a minimum g-extra cut of G;_
2. \(\kappa_{g}(G)\geq g+2\)_,_ \(\kappa_{g+1}(G)\geq\kappa_{g}(G)+g\)_;_
3. \(mc(G-F)\geq N-|F|-(g+1)\) _for any vertex set_ \(F\subseteq V(G)\) _with_ \(|F|\leq\kappa_{g+1}(G)-1\)_;_  
_then, \(t_{g}(G)=\kappa_{g}(G)+g\) under the MM\({}^{\star}\) model._  
Lemma 11: _The vertex number of the maximal independent set \(\alpha(BC_{n,k})=0\) in \(BC_{n,k}\)._  
Proof. : We first prove that \(V(BC_{n,k})-(F_{1}\cup F_{2})\
eq\emptyset\). Next, prove the conclusion in \(V(BC_{n,k})-F_{1}\cup F_{2}\
eq\emptyset\). We prove that \(\alpha(BC_{n,k})=0\) is equivalent to proving that \(V(BC_{n,k})-F_{1}\cup F_{2}\) has no isolated vertices. Suppose there is at least one isolated vertex \(s\) in \(V(BC_{n,k})-F_{1}\cup F_{2}\). Suppose \(F_{1}\subset F_{2}\), since \(F_{2}\) is a g-extra fault set and \(s\) is an isolated vertex; we obtain \(g=0\), which contradicts \(g\geq 1\). Thus, \(F_{1}\
ot\subseteq F_{2}\). Similarly, we have \(F_{2}\
ot\subseteq F_{1}\). Let \(S\) be a maximum independent set of \(BC_{n,k}-F_{1}\cup F_{2}\), and let \(H\) be the induced subgraph by the vertex set \(V(BC_{n,k})-(F_{1}\cup F_{2}\cup S)\). Since \(F_{1}\) and \(F_{2}\) are two indistinguishable g-extra fault sets of \(BC_{n,k}\), there exists one vertex \(u\in F_{2}-F_{1}\) at most, such that \(u\) is connected to \(w\), according to Lemma 1(2). If \(N_{BC_{n,k}}(s)\cap(F_{2}-F_{1})=\emptyset\), then \(s\) is an isolated vertex in \(BC_{n,k}-F_{1}\), which is contradictory to \(g\
eq 0\). Thus, there exists only one vertex \(u\) such that \(u\) and \(s\) are adjacent to each other. Similarly, we infer that there exists only one vertex \(v\in F_{1}-F_{2}\) such that \(v\) is adjacent to \(s\). According to Lemma 2(2), \(\kappa(BC_{n,k})=n+k-1\). We have  
\[ \begin{split}|F_{1}\cap F_{2}|&\geq|N_{BC_{n,k}}(s) \cap(F_{1}\cap F_{2})|\\ &=|N_{BC_{n,k}}(s)|-|N_{BC_{n,k}}(s)\cap(F_{1}-F_{2})|-|N_{BC_{n,k }}(s)\cap(F_{2}-F_{1})|\\ &=n+k-1-2\\ &=n+k-3\end{split}  \tag{5}\]  
Let \(S\) be the set of all isolated vertices. Since \(|F_{2}|=(g+1)k+n\), we have  
\[ \begin{split}\sum_{s\in S}|N_{BC_{n,k}[F_{1}\cap F_{2}]}(s)|& =|S|(n+k-3)\\ &\leq\sum_{v\in F_{1}\cap F_{2}}deg_{BC_{n,k}}(v)\\ &\leq|F_{1}\cap F_{2}|(n+k-1)\\ &\leq(|F_{2}|-1)(n+k-1)\\ &=[(g+1)k+n-2](n+k-1)\end{split}  \tag{6}\]  
Thus, \(|S|\leq\frac{[(g+1)k+n-2](n+k-1)}{n+k-3}\), noting that \(|F_{1}\cup F_{2}|=|F_{1}|+|F_{2}|-|F_{1}\cap F_{2}|\leq 2\kappa_{g}(BC_{n,k})+g\). Let \(H=V(BC_{n,k})-(F_{1}\cup F_{2})-S\). Suppose \(V(H)=\emptyset\); when \(0\leq g\leq n-1\), we have  
\[ \begin{split}|F_{1}\cup F_{2}|+|S|&=|F_{1}|+|F_{2} |-|F_{1}\cap F_{2}|+|S|\\ &\leq 2\kappa_{g}(BC_{n,k})+g+\frac{[(g+1)k+n-2](n+k-1)}{n+k-3}\\ &<(k+1)n^{k+1}\end{split}  \tag{7}\]  
Since \(|V(BC_{n,k})|=(k+1)n^{k+1}\), \(V(BC_{n,k})-(F_{1}\cup F_{2})-S>0\). Hence, \(V(H)\
eq\emptyset\).