## 4 Identifying Subproblem Solutions and Defining the Recursive Formulas
}\Big{\{}\tau^{d,\ell}_{p,0}(x,x_{1})(G _{\alpha_{1}})+\tau^{d,\ell}_{p,0}(x,x_{2})(G_{\alpha_{2}})\Big{\}};\]  
_(3.3.3): If \(C_{T}(G_{\alpha_{2}})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{2}})=\emptyset\), then_  
\[\tau^{d,\ell}_{p,0}(x,y)(G_{\alpha})=\min_{ \begin{subarray}{c}x_{1},x_{2}\in \mathbb{Z}\\ x_{1}+x_{2}=x\\ y\leq x_{1}\leq u_{\max}\\ w_{\min}\leq_{2}\leq x-y\end{subarray} }\Big{\{}\tau^{d,\ell}_{p,0}(x,x_{1}) (G_{\alpha_{1}})+\tau^{d,\ell}_{p,1}(x_{2},x_{2})(G_{\alpha_{2}})\Big{\}}.\]  
**Proof. ** The graph \(G_{\alpha}\) is obtained by connecting every vertex of \(TS(G_{\alpha_{1}})\) to all vertices of \(TS(G_{\alpha_{2}})\), and \(TS(G_{\alpha})=TS(G_{\alpha_{1}})\). By Lemma 4, we have \(C_{T}(G_{\alpha})=\emptyset\) and \(C(G_{\alpha})=C_{\overline{T}}(G_{\alpha})=C_{\overline{T}}(G_{\alpha_{1}}) \cup C_{\overline{T}}(G_{\alpha_{2}})\cup C_{X}(G_{\alpha})\). By Definition 7, Statements (1) and (2) are true. We now consider the following three cases for computing the equation for \(\tau^{d,\ell}_{p,0}(x,y)(G_{\alpha})\). Let \(i=\{1,2\}\). If \(C_{T}(G_{\alpha_{1}})=\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{1}})=\emptyset\), then \(G_{\alpha_{1}}\) has no vertices. Therefore, only three cases need to be considered based on their emptiness:  
Case 1: \(C_{T}(G_{\alpha_{1}})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{1}})\
eq\emptyset\);  
Case 2: \(C_{T}(G_{\alpha_{1}})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{1}})=\emptyset\);  
Case 3: \(C_{T}(G_{\alpha_{1}})=\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{1}})\
eq\emptyset\). Each case above has three subcases. Below, we prove the statements for Case 1 and its subcases. The statements for the other cases and their subcases can be proven similarly. **Case 1: \(C_{T}(G_{\alpha_{1}})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha_{1}})\
eq\emptyset\). ** Let \(w_{min}=p\cdot\omega(G)\) and \(w_{max}=(p+(\ell-1)\cdot d)\cdot\omega(G)\). **Case 1.1: \(C_{T}(G_{\alpha_{2}})\
eq\emptyset\) 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 \(\Gamma^{d,\ell}_{p,2}(x,x_{1},x_{1})\)-clique transversal function of \(G_{\alpha_{1}}\) and \(f_{2}\) be a \(\Gamma^{d,\ell}_{p,2}(x,x_{2},x_{2})\)-clique transversal function of \(G_{\alpha_{2}}\). By Definition 7, \(f_{1}(C)\geq x_{1}\) for every \(C\in C_{T}(G_{\alpha_{1}})\) or every \(C\in C(G_{\alpha_{2}}[TS(G_{\alpha_{1}})])\). Similarly, \(f_{2}(C)\geq x_{2}\) for every \(C\in C_{T}(G_{\alpha_{2}})\) or 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 \(C\in C_{\overline{T}}(G_{\alpha_{1}})\). Let \(f\) be a function of \(G_{\alpha}\) such that \(f(v)=f_{1}(v)\) for every \(v\in V(G_{\alpha_{1}})\), and \(f(v)=f_{2}(v)\) for every \(v\in V(G_{\alpha_{2}})\). By Lemma 4, \(C_{T}(G_{\alpha})=\emptyset\) 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 \(C\in C_{\overline{T}}(G_{\alpha})\). If \(C\in C_{\overline{T}}(G_{\alpha_{1}})\) or \(C\in C_{\overline{T}}(G_{\alpha_{2}})\), then \(f(C)=f_{1}(C)\geq x\) for every \(C\in C_{\overline{T}}(G_{\alpha_{1}})\), and \(f(C)=f_{2}(C)\geq x\) for every \(C\in C_{\overline{T}}(G_{\alpha_{2}})\). If \(C\in C_{X}(G_{\alpha})\), then there exist two cliques \(C_{1}\) and \(C_{2}\) such that \(C=C_{1}\cup C_{2},C_{1}\in C(G_{\alpha_{1}}[TS(G_{\alpha_{1}})])\), and \(C_{2}\in C(G_{\alpha_{2}}[TS(G_{\alpha_{2}})])\)Therefore, \(f(C)=f(C_{1})+f(C_{2})=f_{1}(C_{1})+f_{2}(C_{2})\geq x_{1}+x_{2}=x\). We obtain \(f(C)\geq x\) for every \(C\in C_{\overline{T}}(G_{\alpha})\). Let \(C\in C(G_{\alpha}[TS(G_{\alpha})])\). In this case, \(TS(G_{\alpha})=TS(G_{\alpha_{1}})\). Thus, \(f(C)=f_{1}(C_{1})\geq x_{1}\geq y\).