## 4 Identifying Subproblem Solutions and Defining the Recursive Formulas
Let \(p\in\{0,-1\}\), and let \(d,\ell,x,y,z\) be integers such that \(d,\ell\in\mathbb{Z}^{+}\) are fixed, and \(p\cdot\omega(G)\leq x,y,z\leq(p+(\ell-1)\cdot d)\cdot\omega(G)\). Then, \(\tau^{d,\ell}_{p,0}(x,y)(G_{\alpha})\), \(\tau^{d,\ell}_{p,1}(x,y)(G_{\alpha})\), and \(\tau^{d,\ell}_{p,2}(x,y,z)(G_{\alpha})\) can be computed as follows:_  
eq\emptyset\) _and_ \(C_{\overline{T}}(G)=\emptyset\)_;_ 1. \(\tau^{d,\ell}_{p,0}(x,y)(G_{\alpha})=\infty\)_;_ 1. \(\tau^{d,\ell}_{p,2}(x,y,z)(G_{\alpha})=\infty\)_;_ 1. _Let_ \(w=\max\{x,y\}\)_,_ \(w_{min}=p\cdot\omega(G)\)_, and_ \(w_{max}=(p+(\ell-1)\cdot d)\cdot\omega(G)\)_. Then,_ \(\tau^{d,\ell}_{p,1}(x,y)(G_{\alpha})=\tau^{d,\ell}_{p,1}(w,w)(G_{\alpha})\) _and_ 2. _Assume that_ \(C_{T}(G)\
eq\emptyset\) _and_ \(C_{\overline{T}}(G)\
eq\emptyset\)_;_ 2. \(\tau^{d,\ell}_{p,0}(x,y)(G_{\alpha})=\infty\)_;_ 2. \(t\in\{1,2\}\)_. Let_ \(w=\max\{x,y\}\)_,_ \(w_{min}=p\cdot\omega(G)\)_, and_ \(w_{max}=(p+(\ell-1)\cdot d)\cdot\omega(G)\)_. Then,_ \(\tau^{d,\ell}_{p,2}(x,y,z)(G_{\alpha})=\tau^{d,\ell}_{p,2}(x,w,w)(G_{\alpha})\)_._ \(\tau^{d,\ell}_{p,2}(x,w,w)(G_{\alpha})\) _can be computed as follows:_ 3. _If_ \(C_{T}(G_{\alpha_{1}})\
eq\emptyset\) _and_ \(C_{\overline{T}}(G_{\alpha_{1}})\
eq\emptyset\)_, and_ \(C_{T}(G_{\alpha_{2}})\
eq\emptyset\) _and_ \(C_{\overline{T}}(G_{\alpha_{2}})\
eq\emptyset\)_, then_ 4. _If_ \(C_{T}(G_{\alpha_{i}})\
eq\emptyset\) _and_ \(C_{\overline{T}}(G_{\alpha_{i}})\
eq\emptyset\)_, and_ \(C_{T}(G_{\alpha_{3-i}})\
eq\emptyset\) _and_ \(C_{\overline{T}}(G_{\alpha_{3-i}})=\emptyset\)_, then_ 5. _If_ \(C_{T}(G_{\alpha_{i}})\
eq\emptyset\) _and_ \(C_{\overline{T}}(G_{\alpha_{i}})\
eq\emptyset\)_, and_ \(C_{T}(G_{\alpha_{3-i}})\
eq\emptyset\) _and_ \(C_{\overline{T}}(G_{\alpha_{3-i}})=\emptyset\)_, then_ 6. _If_ \(C_{T}(G_{\alpha_{i}})\
eq\emptyset\) _and_ \(C_{\overline{T}}(G_{\alpha_{i}})=\emptyset\)_, and_ \(C_{T}(G_{\alpha_{3-i}})=\emptyset\) _and_ \(C_{\overline{T}}(G_{\alpha_{3-i}})=\emptyset\)_, then_ 7. _If_ \(C_{T}(G_{\alpha_{i}})\
eq\emptyset\) _and_ \(C_{\overline{T}}(G_{\alpha_{i}})=\emptyset\)_, and_ \(C_{T}(G_{\alpha_{3-i}})=\emptyset\) _and_ \(C_{\overline{T}}(G_{\alpha_{3-i}})=\emptyset\)_, then_ 8. _If_ \(C_{T}(G_{\alpha_{i}})\
eq\emptyset\) _and_ \(C_{\overline{T}}(G_{\alpha_{i}})=\emptyset\)_, and_ \(C_{T}(G_{\alpha_{3-i}})=\emptyset\) _and_ \(C_{\overline{T}}(G_{\alpha_{3-i}})=\emptyset\)_, then_ 9. _If_ \(C_{T}(G_{\alpha_{i}})\
eq\emptyset\) _and_ \(C_{\overline{T}}(G_{\alpha_{i}})=\emptyset\)_, and_ \(C_{T}(G_{\alpha_{3-i}})=\emptyset\) _and_ \(C_{\overline{T}}(G_{\alpha_{3-i}})\
eq\emptyset\)_, then_ 10. \(\tau^{d,\ell}_{p,2}(x,w,w)=\min\limits_{ \begin{subarray}{c}i\in\{1,2\}\\ w_{1},w_{2}\in\mathbb{Z}\\ w_{min}\leq\
u_{1},w_{2}\leq w_{max}\end{subarray} }\left\{\tau^{d,\ell}_{p,1}(w_ {1},w_{1})(G_{\alpha_{i}})+\tau^{d,\ell}_{p,0}(x,w_{2})(G_{\alpha_{3-i}}) \right\}\)_._  
Proof. : By Theorem 1, \(G_{\alpha}\) is a distance-hereditary graph formed by connecting every vertex of \(TS(G_{\alpha_{1}})\) to all vertices of \(TS(G_{\alpha_{2}})\), and \(TS(G_{\alpha})=TS(G_{\alpha_{1}})\cup TS(G_{\alpha_{2}})\). The union of a maximal clique of \(G_{\alpha_{1}}[TS(G_{\alpha_{1}})]\) and a maximal clique of \(G_{\alpha_{1}}[TS(G_{\alpha_{1}})]\) is a maximal clique in \(G_{\alpha}\). Furthermore, \(C_{T}(G_{\alpha})=C(G_{\alpha}[TS(G_{\alpha})])=C_{X}(G_{\alpha})\) by Lemma 3. Therefore, \(C_{T}(G_{\alpha})\
eq\emptyset\). There are two cases to consider:  
* \(C_{T}(G_{\alpha})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha})=\emptyset\)\(\bullet\)\(C_{T}(G_{\alpha})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha})\
eq\emptyset\). **(1) Assume that \(C_{T}(G_{\alpha})\
eq\emptyset\) and \(C_{\overline{T}}(G_{\alpha})=\emptyset\). ** By Definition 7, Statements (1.1) and (1.2) are true. Let \(f\) be a \(\Gamma_{p,1}^{d,\ell}(x,y)\)-clique transversal function of \(G_{\alpha}\), and let \(w=\max\{x,y\}\).