## 4 Identifying Subproblem Solutions and Defining the Recursive Formulas
The function \(f\) is a \(\Gamma_{p,0}^{d,\ell}(x,y)\)-clique transversal function of \(G_{\alpha}\). Consequently,  
\[\tau_{p,0}^{d,\ell}(x,y)(G_{\alpha})\leq\tau_{p,2}^{d,\ell}(x,x_{1},x_{1})(G_{ \alpha_{1}})+\tau_{p,2}^{d,\ell}(x,x_{2},x_{2})(G_{\alpha_{2}}).\]  
Conversely, let \(f\) be a \(\Gamma_{p,0}^{d,\ell}(x,y)\)-clique transversal function of \(G_{\alpha}\). Let \(f_{1}\) be a function of \(G_{\alpha_{1}}\) such that \(f_{1}(v)=f(v)\) for every \(v\in V(G_{\alpha_{1}})\), and let \(f_{2}\) be a function of \(G_{\alpha_{2}}\) such that \(f_{2}(v)=f(v)\) for every \(v\in V(G_{\alpha_{2}})\). By Lemma 4, \(C_{\overline{T}}(G_{\alpha})=\emptyset\) and \(C_{\overline{T}}(G_{\alpha})=C_{\overline{T}}(G_{\alpha_{1}})\cup C_{\overline {T}}(G_{\alpha_{2}})\cup C_{X}(G_{\alpha})\). Let \(C\in C_{\overline{T}}(G_{\alpha_{1}})\cup C_{\overline{T}}(G_{\alpha_{2}})\). Then, \(C\in C_{\overline{T}}\). Therefore, \(f_{1}(C)=f(C)\geq x\) for every \(C\in C_{\overline{T}}(G_{\alpha_{1}})\), and \(f_{2}(C)=f(C)\geq x\) for every \(C_{\overline{T}}(G_{\alpha_{2}})\). Let \(C_{1}\in C(G_{\alpha_{1}}[TS(G_{\alpha_{1}})])\) and \(C_{2}\in C(G_{\alpha_{2}}[TS(G_{\alpha_{2}})])\). The clique \(C=C_{1}\cup C_{2}\) is a maximal clique in \(C_{\overline{T}}(G_{\alpha})\). We have  
\[f_{1}(C_{1})+f_{2}(C_{2})=f(C_{1})+f(C_{2})=f(C)\geq x.\]  
In this case, \(TS(G_{\alpha})=TS(G_{\alpha_{1}})\). Then, \(C_{1}\in C(G_{\alpha}[TS(G_{\alpha})])\) and \(f_{1}(C_{1})=f(C_{1})\geq y\). It is not difficult to see that \(f_{1}(C_{1})\in\{y,y+1,\ldots,w_{max}\}\) and \(f_{2}(C_{2})\in\{x-y,x-y-1,\ldots,w_{min}\}\). Consequently, there exist two integers \(x_{1}\) and \(x_{2}\) such that  
(i) \(y\leq x_{1}\leq w_{max}\);  
(ii) \(w_{min}\leq x_{2}\leq x-y\);  
(iii) \(x_{1}+x_{2}=x_{i}\);  
(iv) \(f_{1}(C_{1})\geq x_{1}\) for every \(C_{1}\in C(G_{\alpha_{1}}[TS(G_{\alpha_{1}})])\); and  
(v) \(f_{2}(C_{2})\geq x_{2}\) for every \(C_{2}\in C(G_{\alpha_{1}}[TS(G_{\alpha_{1}})])\). Note that \(C_{\overline{T}}(G_{\alpha_{1}})\subseteq C(G_{\alpha_{1}}[TS(G_{\alpha_{1}})])\) and \(C_{\overline{T}}(G_{\alpha_{2}})\subseteq C(G_{\alpha_{2}}[TS(G_{\alpha_{2}})])\). Thus, \(f_{1}(C_{1})\geq x_{1}\) for every \(C_{1}\in C_{\overline{T}}(G_{\alpha_{1}})\), and \(f_{2}(C_{2})\geq x_{2}\) for every \(C_{2}\in C_{\overline{T}}(G_{\alpha_{2}})\). The function \(f_{1}\) is a \(\Gamma_{p,2}^{d,\ell}(x,x_{1},x_{1})\)-clique transversal function of \(G_{\alpha_{1}}\), and \(f_{2}\) is a \(\Gamma_{p,2}^{d,\ell}(x,x_{2},x_{2})\)-clique transversal function of \(G_{\alpha_{2}}\). We obtain  
\[\tau_{p,2}^{d,\ell}(x,x_{1},x_{1})(G_{\alpha_{1}})+\tau_{p,2}^{d,\ell}(x,x_{2},x_{2})(G_{\alpha_{2}})\leq\tau_{p,0}^{d,\ell}(x,y)(G_{\alpha}).\]  
Consequently, \(\tau_{p,0}^{d,\ell}(x,y)(G_{\alpha})=\tau_{p,2}^{d,\ell}(x,x_{1},x_{1})(G_{ \alpha_{1}})+\tau_{p,2}^{d,\ell}(x,x_{2},x_{2})(G_{\alpha_{2}})\). Statement (3.1.1) therefore holds. **Case 1.2: \(C_{\overline{T}}(G_{\alpha_{2}})=\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{2}})\
eq\emptyset\). ** Let \(x_{1}\) and \(x_{2}\) be integers such that \(x_{1}+x_{2}=x\), \(y\leq x_{1}\leq w_{max}\), and \(w_{min}\leq x_{2}\leq x-y\). Let \(f_{1}\) be a \(\Gamma_{p,2}^{d,\ell}(x,x_{1},x_{1})\)-clique transversal function of \(G_{\alpha_{1}}\) and \(f_{2}\) be a \(\Gamma_{p,0}^{d,\ell}(x,x_{2})\)-clique transversal function of \(G_{\alpha_{2}}\). By Definition 7, \(f_{1}(C)\geq x_{1}\) for every \(C\in C_{\overline{T}}(G_{\alpha_{1}})\) or every \(C\in C(G_{\alpha_{1}}[TS(G_{\alpha_{1}})])\), and \(f_{2}(C)\geq x_{2}\) for every \(C\in C(G_{\alpha_{2}}[TS(G_{\alpha_{2}})])\). Furthermore, \(f_{1}(C)\geq x\) for every \(C\in C_{\overline{T}}(G_{\alpha_{1}})\) and \(f_{2}(C)\geq x\) for every  
\(f(C)=f_{1}(C_{1})\geq x_{1}\geq y\). The function \(f\) is a \(\Gamma_{p,0}^{d,\ell}(x,y)\)-clique transversal function of \(G_{\alpha}\).