## Appendix A All Constraints in an MILP Formulation for Rank-2 Chemical Graphs
### Applicability Domain
**AD constraints in \(\mathcal{C}_{1}\):**  
**constants:**  
Integers \(\mathrm{cs}^{*}\geq 3\) and \(\mathrm{ch}^{*}\geq 1\); An integer \(d_{\max}\in\{3,4\}\);  
An integer \(n^{*}\in[\mathrm{cs}^{*}+1,\mathrm{cs}^{*}\cdot(d_{\max}-1)^{\mathrm{ch}^{*}}]\);  
**variables for descriptors in \(\mathtt{x}\):**  
A real variable \(\kappa_{1}\geq 0\): \(\kappa_{1}\) represents \(\kappa_{1}(H)\);  
\(\mathrm{dg}(i)\in[0,n^{*}]\) (\(i\in[1,4]\)): \(\mathrm{dg}(i)\) represents the number of vertices of degree \(i\) in \(H\);  
Mass \(\in\mathbb{Z}\): Mass represents \(\sum_{v\in V}\) mass\({}^{*}(\alpha(v))\);  
\(\mathrm{ce}^{\mathrm{co}}(\mathtt{a})\in[0,n^{*}]\), \(\mathtt{a}\in\Lambda\): \(\mathrm{ce}^{\mathrm{co}}(\mathtt{a})\) represents the number of vertices of chemical element  
a in the core of \(H\);  
\(\mathrm{ce}^{\mathrm{nc}}(\mathtt{a})\in[0,n^{*}]\), \(\mathtt{a}\in\Lambda\): \(\mathrm{ce}^{\mathrm{nc}}(\mathtt{a})\) represents the number of vertices of chemical element  
a in the non-core of \(H\);  
\(\mathrm{b}^{\mathrm{co}}(k)\in[0,2n^{*}]\), \(k\in[1,3]\): \(\mathrm{b}^{\mathrm{co}}(k)\) represents the number of \(k\)-bonds in the core of \(H\);  
\(\mathrm{b}^{\mathrm{nc}}(k)\in[0,2n^{*}]\), \(k\in[1,3]\): \(\mathrm{b}^{\mathrm{nc}}(k)\) represents the number of \(k\)-bonds in the non-core of \(H\);  
\(\mathrm{ac}^{\mathrm{co}}(\gamma)\in[0,n^{*}]\), \(\gamma\in\Gamma_{<}\cup\Gamma_{=}\): \(\mathrm{ac}^{\mathrm{co}}(\gamma)\) represents the number of core edges  
in \(H\) that are assigned tuple \(\gamma\in\Gamma_{<}\);  
\(\mathrm{ac}^{\mathrm{nc}}(\gamma)\in[0,n^{*}]\), \(\gamma\in\Gamma_{<}\cup\Gamma_{=}\): \(\mathrm{ac}^{\mathrm{nc}}(\gamma)\) represents the number of non-core edges in  
\(H\) that are assigned tuple \(\gamma\in\Gamma_{<}\);constraints:**  
\[n^{*}\min_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\frac{\kappa_ {1}(G)}{n(G)}\leq\kappa_{1}\leq n^{*}\max_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{ \bar{y}}}\frac{\kappa_{1}(G)}{n(G)},\] (A1) \[n^{*}\min_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\frac{ \text{dg}_{\bar{y}}(G)}{n(G)}\leq\text{dg}(i)\leq n^{*}\max_{G\in D_{n}\cup \text{DB}^{(n^{*})}_{\bar{y}}}\frac{\text{dg}_{\bar{y}}(G)}{n(G)}, i\in[1,4],\] (A2) \[n^{*}\min_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\overline {\text{ms}}(G)\leq\text{Mass}\leq n^{*}\max_{G\in D_{n}\cup\text{DB}^{(n^{*})}_ {\bar{y}}}\overline{\text{ms}}(G),\] (A3) \[n^{*}\min_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\frac{ \text{ce}^{\text{co}}_{\bar{y}}(G)}{n(G)}\leq\text{ce}^{\text{co}}(\mathbf{a })\leq n^{*}\max_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\frac{\text{ce}^ {\text{co}}_{\bar{y}}(G)}{n(G)}, \mathbf{a}\in\Lambda,\] (A4) \[n^{*}\min_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\frac{ \text{ce}^{\text{co}}_{\bar{y}}(G)}{n(G)}\leq\text{ce}^{\text{nc}}(\mathbf{a })\leq n^{*}\max_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\frac{\text{ce}^ {\text{nc}}_{\bar{y}}(G)}{n(G)}, \mathbf{a}\in\Lambda,\] (A5) \[(n^{*}+1)\min_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\frac {b_{\bar{y}}^{\text{co}}(G)}{n(G)+1}\leq\text{be}^{\text{co}}(k)\leq(n^{*}+1) \max_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\frac{b_{\bar{y}}^{\text{co} }(G)}{n(G)+1}, k\in[2,3],\] (A6) \[(n^{*}+1)\min_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\frac {b_{\bar{y}}^{\text{co}}(G)}{n(G)+1}\leq\text{be}^{\text{nc}}(k)\leq(n^{*}+1) \max_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\frac{b_{\bar{y}}^{\text{nc} }(G)}{n(G)+1}, k\in[2,3],\] (A7) \[(n^{*}+1)\min_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\frac {\text{ac}^{\text{co}}_{\bar{y}}(G)}{n(G)+1}\leq\text{ac}^{\text{co}}(\gamma) \leq(n^{*}+1)\max_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\frac{\text{ ac}^{\text{co}}_{\bar{y}}(G)}{n(G)+1}, \gamma\in\Gamma,\] (A8) \[(n^{*}+1)\min_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\frac {\text{ac}^{\text{nc}}_{\bar{y}}(G)}{n(G)+1}\leq\text{ac}^{\text{nc}}(\gamma) \leq(n^{*}+1)\max_{G\in D_{n}\cup\text{DB}^{(n^{*})}_{\bar{y}}}\frac{\text{ ac}^{\text{nc}}_{\bar{y}}(G)}{n(G)+1}, \gamma\in\Gamma.\] (A9)  
In the following, we derive an MILP \(\mathcal{M}(x,g;\mathcal{C}_{2})\) that satisfies the condition in Theorem 2.