## Appendix A All Constraints in an MILP Formulation for Rank-2 Chemical Graphs
### Descriptor for the Number of Adjacency-Configurations
**constants:**  
A set \(\Gamma=\Gamma_{<}\cup\Gamma_{=}\cup\Gamma_{>}\) of proper tuples \((\mathtt{a},\mathtt{b},k)\in\Lambda\times\Lambda\times[1,3]\);  
The set \(\Gamma_{0}=\{(\mathtt{a},\mathtt{b},0)\mid\mathtt{a},\mathtt{b}\in\Lambda \cup\{\epsilon\}\}\);  
**variables:**  
\(\delta_{\tau}(i,\gamma)\in\{0,1\}\), \(i\in E_{1}\cup E_{3}\), \(\gamma\in\Gamma\cup\Gamma_{0}\):  
\(\delta_{\tau}(i,\gamma)=1\Leftrightarrow\) edge \(a_{i}\) is assigned tuple \(\gamma\); i.e., \(\gamma=(\widetilde{\mathtt{a}}(\mathrm{tail}(i),1),\widetilde{\mathtt{a}}( \mathrm{head}(i),1),\widetilde{\beta}(i))\);  
\(\delta_{\tau}(t,1,\gamma)\in\{0,1\}\), \(t\in[2,t^{*}]\), \(\gamma\in\Gamma\cup\Gamma_{0}\):  
\(\delta_{\tau}(t,1,\gamma)=1\Leftrightarrow\) edge \(e_{t}\) is assigned tuple \(\gamma\); i.e., \(\gamma=(\widetilde{\mathtt{a}}(s^{*}+t-1,1),\widetilde{\mathtt{a}}(s^{*}+t,1), \widetilde{\beta}(t,1))\);  
\(\delta_{\tau}(t,i,\gamma)\in\{0,1\}\), \(p\in[1,\mathrm{cs}^{*}]\), \(i\in[2,n_{\text{tree}}]\), \(\gamma\in\Gamma\cup\Gamma_{0}\):  
\(\delta_{\tau}(t,i,\gamma)=1\Leftrightarrow\) edge \(e^{\prime}_{p,i}\), \(p\leq s^{*}\) (or \(e_{p-s^{*},i}\), \(p>s^{*}\)) is assigned tuple \(\gamma\); i.e.,  
\(\gamma=(\widetilde{\mathtt{a}}(p,\text{prt}(i)),\widetilde{\mathtt{a}}(p,i), \widetilde{\beta}(p,i))\);  
\(\delta_{\tau}(s,t,\gamma)\in\{0,1\}\), \(s\in[1,s^{*}]\), \(t\in[1,t^{*}]\), \(\gamma\in\Gamma\cup\Gamma_{0}\):  
\(\delta_{\tau}(s,t,\gamma)=1\Leftrightarrow\) edge \(u_{s,1}v_{t,1}\) is assigned tuple \(\gamma\); i.e., \(\gamma=(\widetilde{\mathtt{a}}(s,1),\widetilde{\mathtt{a}}(s^{*}+t,1), \widetilde{\beta}(s,t))\);  
\(\mathrm{ac}^{\text{co}}(\gamma)\in[0,n^{*}]\), \(\gamma\in\Gamma_{<}\cup\Gamma_{=}\);  
\(\mathrm{ac}^{\text{nc}}(\gamma)\in[0,n^{*}]\), \(\gamma\in\Gamma_{<}\cup\Gamma_{=}\);constraints:**  
\[\sum_{\gamma\in\Gamma\cup\Gamma_{0}}\delta_{\tau}(i,\gamma)=1, i\in E_{1}\cup E_{3}, \tag{100}\] \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{a}] \delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{tail}(i),1), i\in E_{1}\cup E_{3},\] (101) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{b} ]\delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{head}(i),1), i\in E_{1}\cup E_{3},\] (102) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}k\cdot \delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\beta}(i), i\in E_{1}\cup E_{3},\] (103) \[\sum_{\gamma\in\Gamma\cup\Gamma_{0}}\delta_{\tau}(t,1,\gamma)=1, t\in[2,t^{*}],\] (104) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{a }]\delta_{\tau}(t,1,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(s^{*}+t-1, 1), t\in[2,t^{*}],\] (105) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{b }]\delta_{\tau}(t,1,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(s^{*}+t,1), t\in[2,t^{*}],\] (106) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}k\cdot \delta_{\tau}(t,1,(\mathbf{a},\mathbf{b},k))=\widetilde{\beta}(t,1), t\in[2,t^{*}], \tag{107}\]  
\[\sum_{\gamma\in\Gamma\cup\Gamma_{0}}\delta_{\tau}(p,i,\gamma)=1, p\in[1,\text{cs}^{*}],i\in[2,n_{\text{tree}}], \tag{108}\] \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{a }]\delta_{\tau}(p,i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(p,\text{prt }(i)), p\in[1,\text{cs}^{*}],i\in[2,n_{\text{tree}}],\] (109) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{a}] \delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{tail}(i),1), i\in E_{1}\cup E_{3},\]0 (110) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{a}] \delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{tail}(i),1), i\in E_{1}\cup E_{3},\]1 (111) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{a}] \delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{tail}(i),1), i\in E_{1}\cup E_{3},\]2 (112) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{a}] \delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{tail}(i),1), i\in E_{1}\cup E_{3},\]3 (113) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{a}] \delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{tail}(i),1), i\in E_{1}\cup E_{3},\]4 (114) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{a}] \delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{tail}(i),1), i\in E_{1}\cup E_{3},\]5  
\[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{a}] \delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{tail}(i),1), i\in E_{1}\cup E_{3},\]6 \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{a}] \delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{tail}(i),1), i\in E_{1}\cup E_{3},\]7 \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{a}] \delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{tail}(i),1), i\in E_{1}\cup E_{3},\]8 \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{a}] \delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{tail}(i),1), i\in E_{1}\cup E_{3},\]9 \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{b} ]\delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{head}(i),1), i\in E_{1}\cup E_{3},\]0 (117) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{b} ]\delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{head}(i),1), i\in E_{1}\cup E_{3},\]1 (118) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{b} ]\delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{head}(i),1), i\in E_{1}\cup E_{3},\]2  
\[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{b} ]\delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{head}(i),1), i\in E_{1}\cup E_{3},\]3 \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{b} ]\delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{head}(i),1), i\in E_{1}\cup E_{3},\]4 (111) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{b} ]\delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{head}(i),1), i\in E_{1}\cup E_{3},\]5 (112) \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{b} ]\delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{head}(i),1), i\in E_{1}\cup E_{3},\]6 \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{b} ]\delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{head}(i),1), i\in E_{1}\cup E_{3},\]7 \[\sum_{(\mathbf{a},\mathbf{b},k)\in\Gamma\cup\Gamma_{0}}[\mathbf{b} ]\delta_{\tau}(i,(\mathbf{a},\mathbf{b},k))=\widetilde{\alpha}(\text{head}(i),1), i\in E_{1}\cup E_{3},\]8 \[\qquad\qquad+\sum_{