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# networkx.algorithms.connectivity.edge_kcomponents.EdgeComponentAuxGraph

class EdgeComponentAuxGraph

  A simple algorithm to find all k-edge-connected components in a graph.

  Constructing the auxiliary graph (which may take some time) allows for the k-edge-ccs to be found in linear time for arbitrary k.

  Notes

  This implementation is based on [1]. The idea is to construct an auxiliary graph from which the k-edge-ccs can be extracted in linear time. The auxiliary graph is constructed in \(O(|V|\cdot F)\) operations, where F is the complexity of max flow. Querying the components takes an additional \(O(|V|)\) operations. This algorithm can be slow for large graphs, but it handles an arbitrary k and works for both directed and undirected inputs.

  The undirected case for k=1 is exactly connected components. The undirected case for k=2 is exactly bridge connected components. The directed case for k=1 is exactly strongly connected components.

  References

  [1]

  Wang, Tianhao, et al. (2015) A simple algorithm for finding all k-edge-connected components. http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0136264

  Examples

      >>> import itertools as it
      >>> from networkx.utils import pairwise
      >>> from networkx.algorithms.connectivity import EdgeComponentAuxGraph
      >>> # Build an interesting graph with multiple levels of k-edge-ccs
      >>> paths = [
      ...     (1, 2, 3, 4, 1, 3, 4, 2),  # a 3-edge-cc (a 4 clique)
      ...     (5, 6, 7, 5),  # a 2-edge-cc (a 3 clique)
      ...     (1, 5),  # combine first two ccs into a 1-edge-cc
      ...     (0,),  # add an additional disconnected 1-edge-cc
      ... ]
      >>> G = nx.Graph()
      >>> G.add_nodes_from(it.chain(*paths))
      >>> G.add_edges_from(it.chain(*[pairwise(path) for path in paths]))
      >>> # Constructing the AuxGraph takes about O(n ** 4)
      >>> aux_graph = EdgeComponentAuxGraph.construct(G)
      >>> # Once constructed, querying takes O(n)
      >>> sorted(map(sorted, aux_graph.k_edge_components(k=1)))
      [[0], [1, 2, 3, 4, 5, 6, 7]]
      >>> sorted(map(sorted, aux_graph.k_edge_components(k=2)))
      [[0], [1, 2, 3, 4], [5, 6, 7]]
      >>> sorted(map(sorted, aux_graph.k_edge_components(k=3)))
      [[0], [1, 2, 3, 4], [5], [6], [7]]
      >>> sorted(map(sorted, aux_graph.k_edge_components(k=4)))
      [[0], [1], [2], [3], [4], [5], [6], [7]]

  The auxiliary graph is primarily used for k-edge-ccs but it can also speed up the queries of k-edge-subgraphs by refining the search space.

      >>> import itertools as it
      >>> from networkx.utils import pairwise
      >>> from networkx.algorithms.connectivity import EdgeComponentAuxGraph
      >>> paths = [
      ...     (1, 2, 4, 3, 1, 4),
      ... ]
      >>> G = nx.Graph()
      >>> G.add_nodes_from(it.chain(*paths))
      >>> G.add_edges_from(it.chain(*[pairwise(path) for path in paths]))
      >>> aux_graph = EdgeComponentAuxGraph.construct(G)
      >>> sorted(map(sorted, aux_graph.k_edge_subgraphs(k=3)))
      [[1], [2], [3], [4]]
      >>> sorted(map(sorted, aux_graph.k_edge_components(k=3)))
      [[1, 4], [2], [3]]

  __init__(*args, **kwargs)

  Methods

  `construct`(G) | Builds an auxiliary graph encoding edge-connectivity between nodes.
  `k_edge_components`(k) | Queries the auxiliary graph for k-edge-connected components.
  `k_edge_subgraphs`(k) | Queries the auxiliary graph for k-edge-connected subgraphs.