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# Source code for networkx.algorithms.connectivity.connectivity

    """
    Flow based connectivity algorithms
    """

    import itertools
    from operator import itemgetter

    import networkx as nx

    # Define the default maximum flow function to use in all flow based
    # connectivity algorithms.
    from networkx.algorithms.flow import (
        boykov_kolmogorov,
        build_residual_network,
        dinitz,
        edmonds_karp,
        preflow_push,
        shortest_augmenting_path,
    )

    default_flow_func = edmonds_karp

    from .utils import build_auxiliary_edge_connectivity, build_auxiliary_node_connectivity

    __all__ = [
        "average_node_connectivity",
        "local_node_connectivity",
        "node_connectivity",
        "local_edge_connectivity",
        "edge_connectivity",
        "all_pairs_node_connectivity",
    ]

    @nx._dispatchable(graphs={"G": 0, "auxiliary?": 4}, preserve_graph_attrs={"auxiliary"})
    def local_node_connectivity(
        G, s, t, flow_func=None, auxiliary=None, residual=None, cutoff=None
    ):
        r"""Computes local node connectivity for nodes s and t.

        Local node connectivity for two non adjacent nodes s and t is the
        minimum number of nodes that must be removed (along with their incident
        edges) to disconnect them.

        This is a flow based implementation of node connectivity. We compute the
        maximum flow on an auxiliary digraph build from the original input
        graph (see below for details).

        Parameters
        ----------
        G : NetworkX graph
            Undirected graph

        s : node
            Source node

        t : node
            Target node

        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. 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.

        auxiliary : NetworkX DiGraph
            Auxiliary digraph to compute flow based node 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.

        cutoff : integer, float, or None (default: None)
            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.

        Returns
        -------
        K : integer
            local node connectivity for nodes s and t

        Examples
        --------
        This function is not imported in the base NetworkX namespace, so you
        have to explicitly import it from the connectivity package:

        >>> from networkx.algorithms.connectivity import local_node_connectivity

        We use in this example the platonic icosahedral graph, which has node
        connectivity 5.

        >>> G = nx.icosahedral_graph()
        >>> local_node_connectivity(G, 0, 6)
        5

        If you need to compute local connectivity 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 node connectivity, and the residual
        network for the underlying maximum flow computation.

        Example of how to compute local node connectivity 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_node_connectivity
        >>> H = build_auxiliary_node_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 = dict.fromkeys(G, dict())
        >>> # Reuse the auxiliary digraph and the residual network by passing them
        >>> # as parameters
        >>> for u, v in itertools.combinations(G, 2):
        ...     k = local_node_connectivity(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 node
        connectivity. 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
        >>> local_node_connectivity(G, 0, 6, flow_func=shortest_augmenting_path)
        5

        Notes
        -----
        This is a flow based implementation of node connectivity. We compute the
        maximum flow using, by default, the :meth:`edmonds_karp` algorithm (see:
        :meth:`maximum_flow`) on an auxiliary digraph build from the original
        input graph:

        For an undirected graph G having `n` nodes and `m` edges we derive a
        directed graph H with `2n` nodes and `2m+n` arcs by replacing each
        original node `v` with two nodes `v_A`, `v_B` linked by an (internal)
        arc in H. Then for each edge (`u`, `v`) in G we add two arcs
        (`u_B`, `v_A`) and (`v_B`, `u_A`) in H. Finally we set the attribute
        capacity = 1 for each arc in H [1]_ .

        For a directed graph G having `n` nodes and `m` arcs we derive a
        directed graph H with `2n` nodes and `m+n` arcs by replacing each
        original node `v` with two nodes `v_A`, `v_B` linked by an (internal)
        arc (`v_A`, `v_B`) in H. Then for each arc (`u`, `v`) in G we add one arc
        (`u_B`, `v_A`) in H. Finally we set the attribute capacity = 1 for
        each arc in H.

        This is equal to the local node connectivity because the value of
        a maximum s-t-flow is equal to the capacity of a minimum s-t-cut.

        See also
        --------
        :meth:`local_edge_connectivity`
        :meth:`node_connectivity`
        :meth:`minimum_node_cut`
        :meth:`maximum_flow`
        :meth:`edmonds_karp`
        :meth:`preflow_push`
        :meth:`shortest_augmenting_path`

        References
        ----------
        .. [1] Kammer, Frank and Hanjo Taubig. Graph Connectivity. in Brandes and
            Erlebach, 'Network Analysis: Methodological Foundations', Lecture
            Notes in Computer Science, Volume 3418, Springer-Verlag, 2005.
            http://www.informatik.uni-augsburg.de/thi/personen/kammer/Graph_Connectivity.pdf

        """
        if flow_func is None:
            flow_func = default_flow_func

        if auxiliary is None:
            H = build_auxiliary_node_connectivity(G)
        else:
            H = auxiliary

        mapping = H.graph.get("mapping", None)
        if mapping is None:
            raise nx.NetworkXError("Invalid auxiliary digraph.")

        kwargs = {"flow_func": flow_func, "residual": residual}

        if flow_func is not preflow_push:
            kwargs["cutoff"] = cutoff

        if flow_func is shortest_augmenting_path:
            kwargs["two_phase"] = True

        return nx.maximum_flow_value(H, f"{mapping[s]}B", f"{mapping[t]}A", **kwargs)

    @nx._dispatchable
    def node_connectivity(G, s=None, t=None, flow_func=None):
        r"""Returns node connectivity for a graph or digraph G.

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

        Parameters
        ----------
        G : NetworkX graph
            Undirected 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 :meth:`maximum_flow` for
            details). If flow_func is None, the default maximum flow function
            (:meth:`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
            Node connectivity of G, or local node connectivity if source
            and target are provided.

        Examples
        --------
        >>> # Platonic icosahedral graph is 5-node-connected
        >>> G = nx.icosahedral_graph()
        >>> nx.node_connectivity(G)
        5

        You can use alternative flow algorithms for the underlying maximum
        flow computation. 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
        >>> nx.node_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 node connectivity.

        >>> nx.node_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
        :meth:`local_node_connectivity` for details.

        Notes
        -----
        This is a flow based implementation of node connectivity. The
        algorithm works by solving $O((n-\delta-1+\delta(\delta-1)/2))$
        maximum flow problems on an auxiliary digraph. Where $\delta$
        is the minimum degree of G. For details about the auxiliary
        digraph and the computation of local node connectivity see
        :meth:`local_node_connectivity`. This implementation is based
        on algorithm 11 in [1]_.

        See also
        --------
        :meth:`local_node_connectivity`
        :meth:`edge_connectivity`
        :meth:`maximum_flow`
        :meth:`edmonds_karp`
        :meth:`preflow_push`
        :meth:`shortest_augmenting_path`

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

        """
        if (s is not None and t is None) or (s is None and t is not None):
            raise nx.NetworkXError("Both source and target must be specified.")

        # Local node connectivity
        if s is not None and t is not None:
            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")
            return local_node_connectivity(G, s, t, flow_func=flow_func)

        # Global node connectivity
        if G.is_directed():
            if not nx.is_weakly_connected(G):
                return 0
            iter_func = itertools.permutations
            # It is necessary to consider both predecessors
            # and successors for directed graphs

            def neighbors(v):
                return itertools.chain.from_iterable([G.predecessors(v), G.successors(v)])

        else:
            if not nx.is_connected(G):
                return 0
            iter_func = itertools.combinations
            neighbors = G.neighbors

        # Reuse the auxiliary digraph and the residual network
        H = build_auxiliary_node_connectivity(G)
        R = build_residual_network(H, "capacity")
        kwargs = {"flow_func": flow_func, "auxiliary": H, "residual": R}

        # Pick a node with minimum degree
        # Node connectivity is bounded by degree.
        v, K = min(G.degree(), key=itemgetter(1))
        # compute local node connectivity with all its non-neighbors nodes
        for w in set(G) - set(neighbors(v)) - {v}:
            kwargs["cutoff"] = K
            K = min(K, local_node_connectivity(G, v, w, **kwargs))
        # Also for non adjacent pairs of neighbors of v
        for x, y in iter_func(neighbors(v), 2):
            if y in G[x]:
                continue
            kwargs["cutoff"] = K
            K = min(K, local_node_connectivity(G, x, y, **kwargs))

        return K

    @nx._dispatchable
    def average_node_connectivity(G, flow_func=None):
        r"""Returns the average connectivity of a graph G.

        The average connectivity `\bar{\kappa}` of a graph G is the average
        of local node connectivity over all pairs of nodes of G [1]_ .

        .. math::

            \bar{\kappa}(G) = \frac{\sum_{u,v} \kappa_{G}(u,v)}{{n \choose 2}}

        Parameters
        ----------

        G : NetworkX graph
            Undirected graph

        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. See :meth:`local_node_connectivity`
            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 : float
            Average node connectivity

        See also
        --------
        :meth:`local_node_connectivity`
        :meth:`node_connectivity`
        :meth:`edge_connectivity`
        :meth:`maximum_flow`
        :meth:`edmonds_karp`
        :meth:`preflow_push`
        :meth:`shortest_augmenting_path`

        References
        ----------
        .. [1]  Beineke, L., O. Oellermann, and R. Pippert (2002). The average
                connectivity of a graph. Discrete mathematics 252(1-3), 31-45.
                http://www.sciencedirect.com/science/article/pii/S0012365X01001807

        """
        if G.is_directed():
            iter_func = itertools.permutations
        else:
            iter_func = itertools.combinations

        # Reuse the auxiliary digraph and the residual network
        H = build_auxiliary_node_connectivity(G)
        R = build_residual_network(H, "capacity")
        kwargs = {"flow_func": flow_func, "auxiliary": H, "residual": R}

        num, den = 0, 0
        for u, v in iter_func(G, 2):
            num += local_node_connectivity(G, u, v, **kwargs)
            den += 1

        if den == 0:  # Null Graph
            return 0
        return num / den

    @nx._dispatchable
    def all_pairs_node_connectivity(G, nbunch=None, flow_func=None):
        """Compute node connectivity between all pairs of nodes of G.

        Parameters
        ----------
        G : NetworkX graph
            Undirected graph

        nbunch: container
            Container of nodes. If provided node connectivity will be computed
            only over pairs of nodes in nbunch.

        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. 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
        -------
        all_pairs : dict
            A dictionary with node connectivity between all pairs of nodes
            in G, or in nbunch if provided.

        See also
        --------
        :meth:`local_node_connectivity`
        :meth:`edge_connectivity`
        :meth:`local_edge_connectivity`
        :meth:`maximum_flow`
        :meth:`edmonds_karp`
        :meth:`preflow_push`
        :meth:`shortest_augmenting_path`

        """
        if nbunch is None:
            nbunch = G
        else:
            nbunch = set(nbunch)

        directed = G.is_directed()
        if directed:
            iter_func = itertools.permutations
        else:
            iter_func = itertools.combinations

        all_pairs = {n: {} for n in nbunch}

        # Reuse auxiliary digraph and residual network
        H = build_auxiliary_node_connectivity(G)
        mapping = H.graph["mapping"]
        R = build_residual_network(H, "capacity")
        kwargs = {"flow_func": flow_func, "auxiliary": H, "residual": R}

        for u, v in iter_func(nbunch, 2):
            K = local_node_connectivity(G, u, v, **kwargs)
            all_pairs[u][v] = K
            if not directed:
                all_pairs[v][u] = K

        return all_pairs

    @nx._dispatchable(graphs={"G": 0, "auxiliary?": 4})
    def local_edge_connectivity(
        G, s, t, flow_func=None, auxiliary=None, residual=None, cutoff=None
    ):
        r"""Returns local edge connectivity for nodes s and t in G.

        Local edge connectivity for two nodes s and t is the minimum number
        of edges that must be removed to disconnect them.

        This is a flow based implementation of edge connectivity. We compute the
        maximum flow on an auxiliary digraph build from the original
        network (see below for details). This is equal to the local edge
        connectivity because the value of a maximum s-t-flow is equal to the
        capacity of a minimum s-t-cut (Ford and Fulkerson theorem) [1]_ .

        Parameters
        ----------
        G : NetworkX graph
            Undirected or directed graph

        s : node
            Source node

        t : node
            Target node

        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. 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.

        auxiliary : NetworkX DiGraph
            Auxiliary digraph for computing flow based edge connectivity. 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.

        cutoff : integer, float, or None (default: None)
            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.

        Returns
        -------
        K : integer
            local edge connectivity for nodes s and t.

        Examples
        --------
        This function is not imported in the base NetworkX namespace, so you
        have to explicitly import it from the connectivity package:

        >>> from networkx.algorithms.connectivity import local_edge_connectivity

        We use in this example the platonic icosahedral graph, which has edge
        connectivity 5.

        >>> G = nx.icosahedral_graph()
        >>> local_edge_connectivity(G, 0, 6)
        5

        If you need to compute local connectivity 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 local edge connectivity 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 = dict.fromkeys(G, dict())
        >>> # Reuse the auxiliary digraph and the residual network by passing them
        >>> # as parameters
        >>> for u, v in itertools.combinations(G, 2):
        ...     k = local_edge_connectivity(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
        connectivity. 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
        >>> local_edge_connectivity(G, 0, 6, flow_func=shortest_augmenting_path)
        5

        Notes
        -----
        This is a flow based implementation of edge connectivity. We compute the
        maximum flow using, by default, the :meth:`edmonds_karp` algorithm on an
        auxiliary digraph build from the original input graph:

        If the input graph is undirected, we replace each edge (`u`,`v`) with
        two reciprocal arcs (`u`, `v`) and (`v`, `u`) and then we set the attribute
        'capacity' for each arc to 1. If the input graph is directed we simply
        add the 'capacity' attribute. This is an implementation of algorithm 1
        in [1]_.

        The maximum flow in the auxiliary network is equal to the local edge
        connectivity because the value of a maximum s-t-flow is equal to the
        capacity of a minimum s-t-cut (Ford and Fulkerson theorem).

        See also
        --------
        :meth:`edge_connectivity`
        :meth:`local_node_connectivity`
        :meth:`node_connectivity`
        :meth:`maximum_flow`
        :meth:`edmonds_karp`
        :meth:`preflow_push`
        :meth:`shortest_augmenting_path`

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

        """
        if flow_func is None:
            flow_func = default_flow_func

        if auxiliary is None:
            H = build_auxiliary_edge_connectivity(G)
        else:
            H = auxiliary

        kwargs = {"flow_func": flow_func, "residual": residual}

        if flow_func is not preflow_push:
            kwargs["cutoff"] = cutoff

        if flow_func is shortest_augmenting_path:
            kwargs["two_phase"] = True

        return nx.maximum_flow_value(H, s, t, **kwargs)

    @nx._dispatchable
    def edge_connectivity(G, s=None, t=None, flow_func=None, cutoff=None):
        r"""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 :meth:`maximum_flow` for
            details). If flow_func is None, the default maximum flow function
            (:meth:`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.

        cutoff : integer, float, or None (default: None)
            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.

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

        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
        :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
        >>> 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
        :meth:`local_edge_connectivity` for details.

        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 [1]_ ) and
        computing local maximum flow (see :meth:`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 [1]_ .
        For directed graphs, the algorithm does n calls to the maximum
        flow function. This is an implementation of algorithm 8 in [1]_ .

        See also
        --------
        :meth:`local_edge_connectivity`
        :meth:`local_node_connectivity`
        :meth:`node_connectivity`
        :meth:`maximum_flow`
        :meth:`edmonds_karp`
        :meth:`preflow_push`
        :meth:`shortest_augmenting_path`
        :meth:`k_edge_components`
        :meth:`k_edge_subgraphs`

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

        """
        if (s is not None and t is None) or (s is None and t is not None):
            raise nx.NetworkXError("Both source and target must be specified.")

        # Local edge connectivity
        if s is not None and t is not None:
            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")
            return local_edge_connectivity(G, s, t, flow_func=flow_func, cutoff=cutoff)

        # Global edge connectivity
        # reuse auxiliary digraph and residual network
        H = build_auxiliary_edge_connectivity(G)
        R = build_residual_network(H, "capacity")
        kwargs = {"flow_func": flow_func, "auxiliary": H, "residual": R}

        if G.is_directed():
            # Algorithm 8 in [1]
            if not nx.is_weakly_connected(G):
                return 0

            # initial value for \lambda is minimum degree
            L = min(d for n, d in G.degree())
            nodes = list(G)
            n = len(nodes)

            if cutoff is not None:
                L = min(cutoff, L)

            for i in range(n):
                kwargs["cutoff"] = L
                try:
                    L = min(L, local_edge_connectivity(G, nodes[i], nodes[i + 1], **kwargs))
                except IndexError:  # last node!
                    L = min(L, local_edge_connectivity(G, nodes[i], nodes[0], **kwargs))
            return L
        else:  # undirected
            # Algorithm 6 in [1]
            if not nx.is_connected(G):
                return 0

            # initial value for \lambda is minimum degree
            L = min(d for n, d in G.degree())

            if cutoff is not None:
                L = min(cutoff, L)

            # A dominating set is \lambda-covering
            # We need a dominating set with at least two nodes
            for node in G:
                D = nx.dominating_set(G, start_with=node)
                v = D.pop()
                if D:
                    break
            else:
                # in complete graphs the dominating sets will always be of one node
                # thus we return min degree
                return L

            for w in D:
                kwargs["cutoff"] = L
                L = min(L, local_edge_connectivity(G, v, w, **kwargs))

            return L