Fig. 3

Example for the construction of \({\hat{G}}(C)\) from G (top). The graph G has two non-trivial SCCs (indicated by the white and orange vertices, resp.). In addition, there and two singleton SCCs (purple vertices) from which \({\tilde{G}}({\bar{V}})\) is constructed. The middle panel shows the graphs \({\tilde{G}}(C)\). Each is obtained by adding the artificial source and sink vertices a and b. The artificial source of the second SCC has no incident edge and in the DAG \({\tilde{G}}({\bar{V}})\) the artificial sink b has no incoming edge. These vertices are not shown since only the connected components containing C or \({\bar{V}}\) are of interest. The edges (10, 1), (5, 9) and (6, 9) in G form connections between the SCCs and the DAG, resp. Hence they are replaced by corresponding edges to an artificial source or artificial sink vertex according to Definition 5. The bottom panel shows the graphs \({\hat{G}}(C)\) obtained with the help of DFS searches. The reverse post ordering is shown. In the case of the second SCC, the artificial source a is connected to 11 as described in Corollary 6. The back edges (5, 2), (7, 1), (7, 6) and (10, 11) are then replaced with the corresponding edge to a and from b as prescribed by Definition 6. The tree graphs have the same superbubbles as G