## 4 Identifying Subproblem Solutions and Defining the Recursive Formulas
Then, \(f_{1}(\hat{C}_{1})+f_{2}(\hat{C}_{2})=f(\hat{C}_{1})+f(\hat{C}_{2})=f(\hat{C})=x_ {1}+x_{2}\geq w\). Since \(|\hat{C}_{1}|\) and \(|\hat{C}_{2}|\) are no more than \(\omega(G)\), \(w_{min}\leq x_{1},x_{2}\leq w_{max}\). If \(x_{1}+x_{2}>w\), then there exist two integers \(w_{1}\) and \(w_{2}\) such that  
(i)\(w_{min}\leq w_{1},w_{2}\leq w_{max}\);  
(ii)\(w_{1}+w_{2}=w\);  
(iii)\(w_{1}\leq x_{1}\), and \(w_{2}\leq x_{2}\). If \(x_{1}+x_{2}=w\), let \(w_{1}=x_{1}\) and \(w_{2}=x_{2}\). Therefore, there exist two integer \(w_{1}\) and \(w_{2}\) such that \(w_{1}+w_{2}=w\), \(f_{1}(C_{1})\geq w_{1}\) for every \(C_{1}\in C(G_{\alpha_{1}}[TS(G_{\alpha_{1}})])\), and \(f_{2}(C_{2})\geq w_{2}\) for every \(C_{2}\in C(G_{\alpha_{2}}[TS(G_{\alpha_{2}})])\). Obviously, \(w_{min}\leq w_{1},w_{2}\leq w_{max}\). The function \(f_{1}\) is a \(\Gamma_{p,1}^{d,\ell}(w_{1},w_{1})\)-clique transversal function of \(G_{\alpha_{1}}\), and \(f_{2}\) is a \(\Gamma_{p,1}^{d,\ell}(w_{2},w_{2})\)-clique transversal function of \(G_{\alpha_{2}}\). Consequently,  
\[\tau_{p,1}^{d,\ell}(w_{1},w_{1})(G_{\alpha_{1}})+\tau_{p,1}^{d,\ell}(w_{2},w_{2 })(G_{\alpha_{2}})\leq\tau_{p,1}^{d,\ell}(w,w)(G_{\alpha}).\]  
Hence, \(\tau_{p,1}^{d,\ell}(w,w)(G_{\alpha})=\tau_{p,1}^{d,\ell}(w_{1},w_{1})(G_{ \alpha_{1}})+\tau_{p,1}^{d,\ell}(w_{2},w_{2})(G_{\alpha_{2}})\). Statement (1.3) therefore holds. **(2) Assume that \(C_{T}(G_{\alpha})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha})\
eq\emptyset\). ** By Definition 7, \(\tau_{p,0}^{d,\ell}(x,y)(G_{\alpha})=\infty\) and \(\tau_{p,1}^{d,\ell}(x,y)(G_{\alpha})=\infty\). Statements (2.1) and (2.2) therefore hold. Let \(w=\max\{x,y\}\). Using arguments similar to those for proving \(\tau_{p,1}^{d,\ell}(x,y)(G_{\alpha})=\tau_{p,1}^{d,\ell}(w,w)(G_{\alpha})\) when \(C_{T}(G_{\alpha})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha})=\emptyset\), we can prove \(\tau_{p,2}^{d,\ell}(x,y,z)(G_{\alpha})=\tau_{p,2}^{d,\ell}(x,w,w)(G_{\alpha})\). We now consider the equation for \(\tau_{p,2}^{d,\ell}(x,w,w)(G_{\alpha})\). Note that \(C_{T}(G_{\alpha})=C_{T}(G_{\alpha_{1}})\cup C_{T}(G_{\alpha_{2}})\) and \(C_{\overline{T}}(G_{\alpha})=C_{\overline{T}}(G_{\alpha_{1}})\cup C_{ \overline{T}}(G_{\alpha_{2}})\). Let \(i=\{1,2\}\). Given \(C_{T}(G_{\alpha})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha})\
eq\emptyset\), we consider the following:  
Case 1: \(C_{T}(G_{\alpha_{1}})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{1}})\
eq\emptyset\), and \(C_{T}(G_{\alpha_{2}})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{2}})\
eq\emptyset\);  
Case 2: \(C_{T}(G_{\alpha_{i}})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{i}})\
eq\emptyset\), and \(C_{T}(G_{\alpha_{3-i}})=\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{3-i}})\
eq\emptyset\);  
Case 3: \(C_{T}(G_{\alpha_{i}})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{i}})\
eq\emptyset\), and \(C_{T}(G_{\alpha_{3-i}})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{3-i}})=\emptyset\);  
Case 4: \(C_{T}(G_{\alpha_{i}})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{i}})=\emptyset\), and \(C_{T}(G_{\alpha_{3-i}})=\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{3-i}})\
eq\emptyset\). In the following, we prove the statement for Case 1. The statements for the other cases can be proven similarly. Let \(w_{min}=p\cdot\omega(G)\) and \(w_{max}=(p+(\ell-1)\cdot d)\cdot\omega(G)\). Let \(w_{1}\) and \(w_{2}\) be integers such that \(w_{min}\leq w_{1},w_{2}\leq w_{max}\) and \(w_{1}+w_{2}=w\). Let \(f_{1}\) be a \(\Gamma_{p,2}^{d,\ell}(x,w_{1},w_{1})\)-clique transversal function of \(G_{\alpha_{1}}\) and \(f_{2}\) be a \(\Gamma_{p,2}^{d,\ell}(x,w_{2},w_{2})\)-clique transversal function of \(G_{\alpha_{2}}\). By Definition 7, \(f_{1}(C)\geq w_{1}\) for every \(C\in C_{T}(G_{\alpha_{1}})\) and \(C\in C(G_{\alpha_{1}}[TS(G_{\alpha_{1}})])\). Similarly, \(f_{2}(C)\geq w_{2}\) for every \(C\in C_{T}(G_{\alpha_{2}})\) and \(C\in C(G_{\alpha_{2}}[TS(G_{\alpha_{2}})])\).