code: def edge_disjoint_paths(
      G,
      s,
      t,
      flow_func=None,
      cutoff=None,
      auxiliary=None,
      residual=None,
  ):
      """Returns the edges disjoint paths between source and target.
  
      Edge disjoint paths are paths that do not share any edge. The
      number of edge disjoint paths between source and target is equal
      to their edge connectivity.
  
      Parameters
      ----------
      G : NetworkX graph
  
      s : node
          Source node for the flow.
  
      t : node
          Sink node for the flow.
  
      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 :meth:`maximum_flow` for
          details). If flow_func is None, the default maximum flow function
          (:meth:`edmonds_karp`) is used. The choice of the default function
          may change from version to version and should not be relied on.
          Default value: None.
  
      cutoff : integer or None (default: None)
          Maximum number of paths to yield. If specified, the maximum flow
          algorithm will terminate when the flow value reaches or exceeds the
          cutoff. This only works for flows that support the cutoff parameter
          (most do) and is ignored otherwise.
  
      auxiliary : NetworkX DiGraph
          Auxiliary digraph to compute flow based edge connectivity. It has
          to have a graph attribute called mapping with a dictionary mapping
          node names in G and in the auxiliary digraph. If provided
          it will be reused instead of recreated. Default value: None.
  
      residual : NetworkX DiGraph
          Residual network to compute maximum flow. If provided it will be
          reused instead of recreated. Default value: None.
  
      Returns
      -------
      paths : generator
          A generator of edge independent paths.
  
      Raises
      ------
      NetworkXNoPath
          If there is no path between source and target.
  
      NetworkXError
          If source or target are not in the graph G.
  
      See also
      --------
      :meth:`node_disjoint_paths`
      :meth:`edge_connectivity`
      :meth:`maximum_flow`
      :meth:`edmonds_karp`
      :meth:`preflow_push`
      :meth:`shortest_augmenting_path`
  
      Examples
      --------
      We use in this example the platonic icosahedral graph, which has node
      edge connectivity 5, thus there are 5 edge disjoint paths between any
      pair of nodes.
  
      >>> G = nx.icosahedral_graph()
      >>> len(list(nx.edge_disjoint_paths(G, 0, 6)))
      5
  
  
      If you need to compute edge disjoint paths on several pairs of
      nodes in the same graph, it is recommended that you reuse the
      data structures that NetworkX uses in the computation: the
      auxiliary digraph for edge connectivity, and the residual
      network for the underlying maximum flow computation.
  
      Example of how to compute edge disjoint paths among all pairs of
      nodes of the platonic icosahedral graph reusing the data
      structures.
  
      >>> import itertools
      >>> # You also have to explicitly import the function for
      >>> # building the auxiliary digraph from the connectivity package
      >>> from networkx.algorithms.connectivity import build_auxiliary_edge_connectivity
      >>> H = build_auxiliary_edge_connectivity(G)
      >>> # And the function for building the residual network from the
      >>> # flow package
      >>> from networkx.algorithms.flow import build_residual_network
      >>> # Note that the auxiliary digraph has an edge attribute named capacity
      >>> R = build_residual_network(H, "capacity")
      >>> result = {n: {} for n in G}
      >>> # Reuse the auxiliary digraph and the residual network by passing them
      >>> # as arguments
      >>> for u, v in itertools.combinations(G, 2):
      ...     k = len(list(nx.edge_disjoint_paths(G, u, v, auxiliary=H, residual=R)))
      ...     result[u][v] = k
      >>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2))
      True
  
      You can also use alternative flow algorithms for computing edge disjoint
      paths. For instance, in dense networks the algorithm
      :meth:`shortest_augmenting_path` will usually perform better than
      the default :meth:`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(list(nx.edge_disjoint_paths(G, 0, 6, flow_func=shortest_augmenting_path)))
      5
  
      Notes
      -----
      This is a flow based implementation of edge disjoint paths. We compute
      the maximum flow between source and target on an auxiliary directed
      network. The saturated edges in the residual network after running the
      maximum flow algorithm correspond to edge disjoint paths between source
      and target in the original network. This function handles both directed
      and undirected graphs, and can use all flow algorithms from NetworkX flow
      package.
  
      """
      if s not in G:
          raise nx.NetworkXError(f"node {s} not in graph")
      if t not in G:
          raise nx.NetworkXError(f"node {t} not in graph")
  
      if flow_func is None:
          flow_func = default_flow_func
  
      if auxiliary is None:
          H = build_auxiliary_edge_connectivity(G)
      else:
          H = auxiliary
  
      # Maximum possible edge disjoint paths
      possible = min(H.out_degree(s), H.in_degree(t))
      if not possible:
          raise NetworkXNoPath
  
      if cutoff is None:
          cutoff = possible
      else:
          cutoff = min(cutoff, possible)
  
      # Compute maximum flow between source and target. Flow functions in
      # NetworkX return a residual network.
      kwargs = {
          "capacity": "capacity",
          "residual": residual,
          "cutoff": cutoff,
          "value_only": True,
      }
      if flow_func is preflow_push:
          del kwargs["cutoff"]
      if flow_func is shortest_augmenting_path:
          kwargs["two_phase"] = True
      R = flow_func(H, s, t, **kwargs)
  
      if R.graph["flow_value"] == 0:
          raise NetworkXNoPath
  
      # Saturated edges in the residual network form the edge disjoint paths
      # between source and target
      cutset = [
          (u, v)
          for u, v, d in R.edges(data=True)
          if d["capacity"] == d["flow"] and d["flow"] > 0
      ]
      # This is equivalent of what flow.utils.build_flow_dict returns, but
      # only for the nodes with saturated edges and without reporting 0 flows.
      flow_dict = {n: {} for edge in cutset for n in edge}
      for u, v in cutset:
          flow_dict[u][v] = 1
  
      # Rebuild the edge disjoint paths from the flow dictionary.
      paths_found = 0
      for v in list(flow_dict[s]):
          if paths_found >= cutoff:
              # preflow_push does not support cutoff: we have to
              # keep track of the paths founds and stop at cutoff.
              break
          path = [s]
          if v == t:
              path.append(v)
              yield path
              continue
          u = v
          while u != t:
              path.append(u)
              try:
                  u, _ = flow_dict[u].popitem()
              except KeyError:
                  break
          else:
              path.append(t)
              yield path
              paths_found += 1
  
file_path: networkx/networkx/algorithms/connectivity/disjoint_paths.py
imports: import networkx
  from networkx.algorithms.flow import edmonds_karp, preflow_push, shortest_augmenting_path
  from networkx.exception import NetworkXNoPath
  from itertools import filterfalse
  from .utils import build_auxiliary_edge_connectivity, build_auxiliary_node_connectivity
label_desc: Use image node_id 1 for calling a global function with example usage: edge_disjoint_paths(G, s, t, flow_func, cutoff, auxiliary, residual) without return types