**4.5. Connectivity testing**

For k-connectivity testing in a graph with *n* vertices, we use existing algorithms from the graph theory [9]. The complexity of this algorithm is *<sup>O</sup>*(*<sup>k</sup>* <sup>∗</sup> *<sup>n</sup>*3), under the condition that *<sup>k</sup>* <sup>&</sup>lt; <sup>√</sup>*<sup>n</sup>* which is true in our case.

## **4.6. Graph consolidation**

In this step, the algorithm finds sub-graphs satisfying the connectivity requirements and transforms each subgraph into a single vertex. The formal specification of the graph consolidation step is described by pseudo code in algorithm 2 which is explained in the following list. Figure 7 shows an example of the operation of the graph consolidation step.

	- (a) If it contains special articulation points, then they are removed from the component.
	- (b) All vertices from the component are transformed into a single vertex in the consolidated graph.
	- (c) The consolidated vertex inherits all edges of the original vertices to other vertices in the graph. Other vertices are vertices not belonging to the same biconnected component.
	- (d) Duplicated edges in the consolidated graph are removed.
