## 4 Numerical Examples and Their Analysis
As we can see in Table 3, the proposed algorithm successfully detects and eliminates the cycle and quickly solves the problem, providing an optimal cycle-free attack path, shown in Figure 3(b). For completeness, we consider an unweighted attack graph similar to that presented in Figure 3(a), that is, we set \(\pi_{j}=0,\forall j\in V\) and \(\tau_{kj}=1,\forall(k,j)\in E\). This graph is presented in Figure 4(a). **Example 4**. : _The AND/OR attack shown in Figure 4(a) is an unweighted attack graph with cycles. The intruder's goal is to access node \(7\)._  
The solution for the attack graph in Figure 4(a) is presented in Table 4 and shown in Figure 4(b). Figure 5: (a) The unweighted attack graph with cycles. (b) Extracted minimum-length attack path. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Step**Node** & _Start_ & 2 & 3 & 5 & 6 & 7 \\ \hline Initial & \(t_{S}=0\) & & & \(u_{5}=\infty\) & \(u_{6}=\infty\) & \(u_{7}=\infty\) \\ (1) & & & & \(u_{5}=9.5\) & \(u_{6}=3\) & \(u_{7}=\infty\) \\ (2) & & & & & \(t_{6}=3\) & \\ (3) & & \(t_{3}=7\) & & & & \\ (2) & & & & \(u_{5}=8.5\) & & \(u_{7}=18\) \\ (3) & & & & \(t_{5}=8.5\) & & \\ (1) & & \(t_{2}=10\) & & & & \\ (2) & & & & & & \(u_{7}=12.5\) \\ (3) & & & & & & \(t_{7}=12.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Solution to Example 3. Figure 5b shows that the solution presented in Table 4 leads to two equivalent attack paths with the same shortest length of 3 min. The first extracted path is AP1 = {start \(\rightarrow\)\(6\rightarrow\)\(3\to 7\)) (see the purple path in Figure 5b) and the second one is AP2 = {start \(\rightarrow\)\(5\to 2\to 7\)) (see the red path in Figure 5b). The obtained paths differ from the path found in the weighted attack graph, shown in Figure 4b. The paths presented in Figure 5b are shorter in length and include fewer nodes that the intruder must compromise before the target will be accessed. The attack graphs presented in Examples 3 and 4 (see Figures 4a and 5a, respectively) include cycles that the proposed algorithm successfully processed and eliminated. **Remark 1**. : _Unlike the cycle eliminating algorithms [10; 34], in which the cycle is detected by the decision-maker, the proposed algorithm detects and eliminates the cycle without human intervention._  
**Remark 2**. : _In some practical cases, the problem with cycles may not be solved by the proposed algorithm, in which case the algorithm terminates without accessing the target._  
A numerical example of such a case is presented in Example 5. **Example 5**. : _The AND/OR attack graph shown in Figure 6 is equipped with the times on nodes and arcs and has cycles. The intruder's goal is to access node **7**._  
The execution of the algorithm is presented in Table 5. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Step** & **Node** & **Start** & **2** & **3** & **5** & **6** & **7** \\ \hline Initial & \(t_{S}=0\) & & & & \(u_{5}=\infty\) & \(u_{6}=\infty\) & \(u_{7}=\infty\) \\ (1) & & & & & \(u_{5}=1\) & \(u_{6}=1\) & \(u_{7}=\infty\) \\ (2) & & & & \(t_{5}=1\) & \(t_{6}=1\) & \\ (3) & & & & & & \(u_{7}=3\) \\ (2) & & & & & & \(t_{7}=3\) \\ (3) & & & & & & \(t_{7}=3\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Solution to Example 4. Figure 6: Attack graph with cycles without an attack path. As one can see in Table 5, the algorithm cannot set the permanent time label to an unvisited OR node (target) **7** (see the last row in Table 5), and therefore, must stop before the target is reached. This means that the solution does not exist and, thus, this problem is unsolvable, that is, the target node **7** is impossible to access.