Fig. 4

Two simple cycles having a vertex v of degree two are depicted in the first column. Their sets of corresponding circles obtained by splitting v into \(v_{1}\) and \(v_{2}\) are depicted in the second column. This set is of size 1 for the upper simple cycle containing the gray loop \(\{v,v\}\), and of size 2 for the lower simple cycle. An \(\mathcal {O}\)-scenario for a simple cycle yields a scenario of the same cost and length transforming the graphs in the second column to those that become terminal once \(v_1\) and \(v_2\) are merged. One possible outcome of such a scenario is presented in the third column