• Team 1, presented by ...:

The team checked for coherency of the references and referenced-by fields. Taking the intersection of the forward edges and backward edges they got only 64 edges in the entire graph.

The team surveyed different algorithms for solving the problem. In the litterature the problem is referred to as the Multy-way cutting problem.

Similar works are listed below: According to the same survey, the same problem for directed acyclyc graphs is solvable in polynomail time.

After that the team presented the behaviour of the graph, the graph has the dimention of 9 and satisfies the small world graph property. This is demonstrated by the plot for the indegree edges.

1. Defining a non-fit cluster which contains all the nodes which don't have a meanningful relation with other nodes.
2. Finding clusters inside the graph (Polynomial Algorithm exists), and using the clusters to define the areas.
• Team 2, presented by ...:

The 2nd team considers the problem as an instance of the edge deletion problem. A solution to the edge deletion problem removes a minimal number of edges from the graph to make it satisfy some condition.

A property is called hereditary if it is maintained by subgraphs (e.g., planarity). If the property we seek is hereditary then the edge deletion problem is NP-complete.

The requirements of our problem are not hereditary. However the following problem which is quite similar is:
Having at least n connected components (classes) with at most one anchor in each class.

Having this in mind, we have some intuition why the problem may be NP-complete (and actually, according to [YAN78] it is).

Commodity: pair of nodes where the first is the source and the second the destination.
Given a graph with a set of commodities, our problem consists of finding a partition such that every two pairs are disconnected and the sum of the capacities of the removed edges is minimal (min-multicut problem).

This problem has an associated max-flow problem but it turns out that in that case, the (multicommodity) max-flow is not equal anymore to the min-multicut (as it is for the one-commodity problem). Max-flow is indeed smaller than min-multicut.

In [GVY93], we found algorithms based on linear programming.

• Team 3, presented by ...:

The group observed that in this kind of graphs, there are 2 types of nodes.

1. The nodes that are in the highly connected areas.
2. The nodes that are in the boundires of highly connected areas
In [FLGC02] we find two algorithms for this problem :

1. Exact algortihm : no exact complexity given.
2. Approximate algorithm