## 3 Results
### Results on Phase 2.
\begin{table}
\begin{tabular}{l r r r r r r} \hline \hline \(\pi\) & \(y^{*}\) & \(n^{*}\) & \(|F^{*}|\)/\#I & IP-Time & \#G^{*}\) & **G-Time** \\ \hline K\({}_{\rm row}\) & 5 & 15 & 12/12 & 11.64 & 100 & 1386.7 \\ K\({}_{\rm row}\) & 5 & 20 & 12/12 & 23.84 & 12 & \(>\)l h \\ K\({}_{\rm row}\) & 5 & 25 & 12/12 & 33.71 & 12 & \(>\)l h \\ K\({}_{\rm row}\) & 5 & 30 & 12/12 & 61.85 & 12 & \(>\)l h \\ Mp & 150 & 15 & 12/12 & 9.80 & 100 & 1614.3 \\ Mp & 150 & 20 & 12/12 & 20.15 & 12 & \(>\)l h \\ Mp & 150 & 25 & 12/12 & 36.42 & 12 & \(>\)l h \\ Mp & 150 & 30 & 12/12 & 40.58 & 12 & \(>\)l h \\ Br & 250 & 15 & 11/12 & 10.25 & 100 & 1756.1 \\ Br & 250 & 19 & 12/12 & 16.02 & 12 & \(>\)l h \\ Br & 250 & 22 & 12/12 & 23.63 & 12 & \(>\)l h \\ Br & 250 & 25 & 12/12 & 63.84 & 12 & \(>\)l h \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results of Steps 4 and 5 with \(d_{\rm max}=3\) and \(\theta=2\). \begin{table}
\begin{tabular}{l r r r r r r} \hline \hline \(\pi\) & \(y^{*}\) & \(n^{*}\) & \(|F^{*}|\)/\#I & IP-Time & \#G^{*}\) & **G-Time** \\ \hline K\({}_{\rm row}\) & 5 & 15 & 11/12 & 28.15 & 100 & 20.3 \\ K\({}_{\rm row}\) & 5 & 20 & 12/12 & 71.90 & 12 & \(>\)l h \\ K\({}_{\rm row}\) & 5 & 25 & 12/12 & 112.71 & 12 & \(>\)l h \\ K\({}_{\rm row}\) & 5 & 30 & 12/12 & 267.21 & 12 & \(>\)l h \\ Mp & 150 & 15 & 9/12 & 22.53 & 100 & 2748.1 \\ Mp & 150 & 20 & 11/12 & 53.44 & 11 & \(>\)l h \\ Mp & 150 & 25 & 12/12 & 143.33 & 12 & \(>\)l h \\ Mp & 150 & 30 & 12/12 & 220.63 & 12 & \(>\)l h \\ Br & 250 & 15 & 6/12 & 27.33 & 100 & 254.2 \\ Br & 250 & 19 & 9/12 & 75.50 & 9 & \(>\)l h \\ Br & 250 & 22 & 11/12 & 133.01 & 11 & \(>\)l h \\ Br & 250 & 25 & 12/12 & 228.75 & 12 & \(>\)l h \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results of Steps 4 and 5 with \(d_{\rm max}=4\) and \(\theta=2\). \begin{table}
\begin{tabular}{l r r r r r} \hline \hline \(\pi\) & \(y^{*}\) & \(n^{*}\) & \(|F^{*}|\)/\#I & IP-Time & \#G^{*}\) & **G-Time** \\ \hline K\({}_{\rm row}\) & 5 & 15 & 11/12 & 28.15 & 100 & 20.3 \\ K\({}_{\rm row}\) & 5 & 20 & 12/12 & 71.90 & 12 & \(>\)l h \\ K\({}_{\rm row}\) & 5 & 25 & 12/12 & 112.71 & 12 & \(>\)l h \\ K\({}_{\rm row}\) & 5 & 30 & 12/12 & 267.21 & 12 & \(>\)l h \\ Mp & 150 & 15 & 9/12 & 22.53 & 100 & 2748.1 \\ Mp & 150 & 20 & 11/12 & 53.44 & 11 & \(>\)l h \\ Mp & 150 & 25 & 12/12 & 143.33 & 12 & \(>\)l h \\ Mp & 150 & 30 & 12/12 & 220.63 & 12 & \(>\)l h \\ Br & 250 & 15 & 6/12 & 27.33 & 100 & 254.2 \\ Br & 250 & 19 & 9/12 & 75.50 & 9 & \(>\)l h \\ Br & 250 & 22 & 11/12 & 133.01 & 11 & \(>\)l h \\ Br & 250 & 25 & 12/12 & 228.75 & 12 & \(>\)l h \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Results of Steps 4 and 5 with \(d_{\rm max}=3\) and \(\theta=0\). Figure 12a-c illustrate some rank-2 chemical graphs \(G^{*}\) with \(\theta(G^{*})=2\) constructed from the vector \(g^{*}\) obtained by solving the MILP in Step 4. Step 5.In this step, we modified the algorithms proposed by [PERSON] et al. (2015) and [PERSON] et al. (2016) to enumerate all rank-2 graphs \(G^{*}\in\mathcal{G}\) such that \(f(G^{*})=x^{*}\) for each \(x^{*}\in F^{*}\). We stop the execution when either the total number of graphs inferred over all vectors \(x^{*}\in F^{*}\) exceeds 100 or the execution time exceeds one hour. Tables 3-8 show the results on Step 5 for \(d_{\max}=3\) and 4, respectively,  
* #\(G^{*}\): the number of all (or up to 100) rank-2 chemical graphs \(G^{*}\) that are computed under 1 h time limit in Step 5, where \(f(G^{*})=x^{*}\) for some \(x^{*}\in F^{*}\). (Note that \(|F^{*}|\) such graphs \(G^{*}\) have been found in Step 4, and Figures 10-12 illustrate some of such graphs \(G^{*}\). );
* G-time: the running time (sec.) to execute Step 5, where \"\(>\)l h\" means that the execution time exceeds the limit. We also conducted some additional experiments to demonstrate that our MILP-based method is flexible to control conditions on the inference of chemical graphs.