edge_connectivity¶

edge_connectivity(G, s=None, t=None, flow_func=None)[source]¶

Returns the edge connectivity of the graph or digraph G.

The edge connectivity is equal to the minimum number of edges that must be removed to disconnect G or render it trivial. If source and target nodes are provided, this function returns the local edge connectivity: the minimum number of edges that must be removed to break all paths from source to target in G.

Parameters :

G : NetworkX graph Undirected or directed 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 :

K : integer Edge connectivity for G, or local edge connectivity if source and target were provided

See also

local_edge_connectivity(), local_node_connectivity(), node_connectivity(), maximum_flow(), edmonds_karp(), preflow_push(), shortest_augmenting_path()

Notes

This is a flow based implementation of global edge connectivity. For undirected graphs the algorithm works by finding a ‘small’ dominating set of nodes of G (see algorithm 7 in [R213] ) and computing local maximum flow (see local_edge_connectivity()) between an arbitrary node in the dominating set and the rest of nodes in it. This is an implementation of algorithm 6 in [R213] . For directed graphs, the algorithm does n calls to the maximum flow function. This is an implementation of algorithm 8 in [R213] .

References

[R213]

(1, 2, 3, 4) Abdol-Hossein Esfahanian. Connectivity Algorithms. http://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf

Examples

>>> # Platonic icosahedral graph is 5-edge-connected
>>> G = nx.icosahedral_graph()
>>> nx.edge_connectivity(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
>>> nx.edge_connectivity(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.