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# minimum_edge_cut

minimum_edge_cut(G, s=None, t=None, flow_func=None)

  Returns a set of edges of minimum cardinality that disconnects G.

  If source and target nodes are provided, this function returns the set of edges of minimum cardinality that, if removed, would break all paths among source and target in G. If not, it returns a set of edges of minimum cardinality that disconnects G.

  Parameters:
    G NetworkX graph
    s node

      Source node. Optional. Default value: None.

    t node

      Target node. Optional. Default value: None.

    flow_func function

      A function for computing the maximum flow among a pair of nodes. The function has to accept at least three parameters: a Digraph, a source node, and a target node. And return a residual network that follows NetworkX conventions (see `maximum_flow()` for details). If flow_func is None, the default maximum flow function (`edmonds_karp()`) is used. See below for details. The choice of the default function may change from version to version and should not be relied on. Default value: None.

  Returns:
    cutset set

      Set of edges that, if removed, would disconnect G. If source and target nodes are provided, the set contains the edges that if removed, would destroy all paths between source and target.

  See also

  `minimum_st_edge_cut()`
  `minimum_node_cut()`
  `stoer_wagner()`
  `node_connectivity()`
  `edge_connectivity()`
  `maximum_flow()`
  `edmonds_karp()`
  `preflow_push()`
  `shortest_augmenting_path()`

  Notes

  This is a flow based implementation of minimum edge cut. For undirected graphs the algorithm works by finding a ‘small’ dominating set of nodes of G (see algorithm 7 in [1]) and computing the maximum flow between an arbitrary node in the dominating set and the rest of nodes in it. This is an implementation of algorithm 6 in [1]. For directed graphs, the algorithm does n calls to the max flow function. The function raises an error if the directed graph is not weakly connected and returns an empty set if it is weakly connected. It is an implementation of algorithm 8 in [1].

  References

  [1] (1,2,3)

  Abdol-Hossein Esfahanian. Connectivity Algorithms. http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf

  Examples

      >>> # Platonic icosahedral graph has edge connectivity 5
      >>> G = nx.icosahedral_graph()
      >>> len(nx.minimum_edge_cut(G))
      5

  You can use alternative flow algorithms for the underlying maximum flow computation. In dense networks the algorithm `shortest_augmenting_path()` will usually perform better than the default `edmonds_karp()`, which is faster for sparse networks with highly skewed degree distributions. Alternative flow functions have to be explicitly imported from the flow package.

      >>> from networkx.algorithms.flow import shortest_augmenting_path
      >>> len(nx.minimum_edge_cut(G, flow_func=shortest_augmenting_path))
      5

  If you specify a pair of nodes (source and target) as parameters, this function returns the value of local edge connectivity.

      >>> nx.edge_connectivity(G, 3, 7)
      5

  If you need to perform several local computations among different pairs of nodes on the same graph, it is recommended that you reuse the data structures used in the maximum flow computations. See `local_edge_connectivity()` for details.