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# Source code for networkx.classes.graph

    """Base class for undirected graphs.

    The Graph class allows any hashable object as a node
    and can associate key/value attribute pairs with each undirected edge.

    Self-loops are allowed but multiple edges are not (see MultiGraph).

    For directed graphs see DiGraph and MultiDiGraph.
    """

    from copy import deepcopy
    from functools import cached_property

    import networkx as nx
    from networkx import convert
    from networkx.classes.coreviews import AdjacencyView
    from networkx.classes.reportviews import DegreeView, EdgeView, NodeView
    from networkx.exception import NetworkXError

    __all__ = ["Graph"]

    class _CachedPropertyResetterAdj:
        """Data Descriptor class for _adj that resets ``adj`` cached_property when needed

        This assumes that the ``cached_property`` ``G.adj`` should be reset whenever
        ``G._adj`` is set to a new value.

        This object sits on a class and ensures that any instance of that
        class clears its cached property "adj" whenever the underlying
        instance attribute "_adj" is set to a new object. It only affects
        the set process of the obj._adj attribute. All get/del operations
        act as they normally would.

        For info on Data Descriptors see: https://docs.python.org/3/howto/descriptor.html
        """

        def __set__(self, obj, value):
            od = obj.__dict__
            od["_adj"] = value
            # reset cached properties
            props = ["adj", "edges", "degree"]
            for prop in props:
                if prop in od:
                    del od[prop]

    class _CachedPropertyResetterNode:
        """Data Descriptor class for _node that resets ``nodes`` cached_property when needed

        This assumes that the ``cached_property`` ``G.node`` should be reset whenever
        ``G._node`` is set to a new value.

        This object sits on a class and ensures that any instance of that
        class clears its cached property "nodes" whenever the underlying
        instance attribute "_node" is set to a new object. It only affects
        the set process of the obj._adj attribute. All get/del operations
        act as they normally would.

        For info on Data Descriptors see: https://docs.python.org/3/howto/descriptor.html
        """

        def __set__(self, obj, value):
            od = obj.__dict__
            od["_node"] = value
            # reset cached properties
            if "nodes" in od:
                del od["nodes"]

    class Graph:
        """
        Base class for undirected graphs.

        A Graph stores nodes and edges with optional data, or attributes.

        Graphs hold undirected edges.  Self loops are allowed but multiple
        (parallel) edges are not.

        Nodes can be arbitrary (hashable) Python objects with optional
        key/value attributes, except that `None` is not allowed as a node.

        Edges are represented as links between nodes with optional
        key/value attributes.

        Parameters
        ----------
        incoming_graph_data : input graph (optional, default: None)
            Data to initialize graph. If None (default) an empty
            graph is created.  The data can be any format that is supported
            by the to_networkx_graph() function, currently including edge list,
            dict of dicts, dict of lists, NetworkX graph, 2D NumPy array, SciPy
            sparse matrix, or PyGraphviz graph.

        attr : keyword arguments, optional (default= no attributes)
            Attributes to add to graph as key=value pairs.

        See Also
        --------
        DiGraph
        MultiGraph
        MultiDiGraph

        Examples
        --------
        Create an empty graph structure (a "null graph") with no nodes and
        no edges.

        >>> G = nx.Graph()

        G can be grown in several ways.

        **Nodes:**

        Add one node at a time:

        >>> G.add_node(1)

        Add the nodes from any container (a list, dict, set or
        even the lines from a file or the nodes from another graph).

        >>> G.add_nodes_from([2, 3])
        >>> G.add_nodes_from(range(100, 110))
        >>> H = nx.path_graph(10)
        >>> G.add_nodes_from(H)

        In addition to strings and integers any hashable Python object
        (except None) can represent a node, e.g. a customized node object,
        or even another Graph.

        >>> G.add_node(H)

        **Edges:**

        G can also be grown by adding edges.

        Add one edge,

        >>> G.add_edge(1, 2)

        a list of edges,

        >>> G.add_edges_from([(1, 2), (1, 3)])

        or a collection of edges,

        >>> G.add_edges_from(H.edges)

        If some edges connect nodes not yet in the graph, the nodes
        are added automatically.  There are no errors when adding
        nodes or edges that already exist.

        **Attributes:**

        Each graph, node, and edge can hold key/value attribute pairs
        in an associated attribute dictionary (the keys must be hashable).
        By default these are empty, but can be added or changed using
        add_edge, add_node or direct manipulation of the attribute
        dictionaries named graph, node and edge respectively.

        >>> G = nx.Graph(day="Friday")
        >>> G.graph
        {'day': 'Friday'}

        Add node attributes using add_node(), add_nodes_from() or G.nodes

        >>> G.add_node(1, time="5pm")
        >>> G.add_nodes_from([3], time="2pm")
        >>> G.nodes[1]
        {'time': '5pm'}
        >>> G.nodes[1]["room"] = 714  # node must exist already to use G.nodes
        >>> del G.nodes[1]["room"]  # remove attribute
        >>> list(G.nodes(data=True))
        [(1, {'time': '5pm'}), (3, {'time': '2pm'})]

        Add edge attributes using add_edge(), add_edges_from(), subscript
        notation, or G.edges.

        >>> G.add_edge(1, 2, weight=4.7)
        >>> G.add_edges_from([(3, 4), (4, 5)], color="red")
        >>> G.add_edges_from([(1, 2, {"color": "blue"}), (2, 3, {"weight": 8})])
        >>> G[1][2]["weight"] = 4.7
        >>> G.edges[1, 2]["weight"] = 4

        Warning: we protect the graph data structure by making `G.edges` a
        read-only dict-like structure. However, you can assign to attributes
        in e.g. `G.edges[1, 2]`. Thus, use 2 sets of brackets to add/change
        data attributes: `G.edges[1, 2]['weight'] = 4`
        (For multigraphs: `MG.edges[u, v, key][name] = value`).

        **Shortcuts:**

        Many common graph features allow python syntax to speed reporting.

        >>> 1 in G  # check if node in graph
        True
        >>> [n for n in G if n < 3]  # iterate through nodes
        [1, 2]
        >>> len(G)  # number of nodes in graph
        5

        Often the best way to traverse all edges of a graph is via the neighbors.
        The neighbors are reported as an adjacency-dict `G.adj` or `G.adjacency()`

        >>> for n, nbrsdict in G.adjacency():
        ...     for nbr, eattr in nbrsdict.items():
        ...         if "weight" in eattr:
        ...             # Do something useful with the edges
        ...             pass

        But the edges() method is often more convenient:

        >>> for u, v, weight in G.edges.data("weight"):
        ...     if weight is not None:
        ...         # Do something useful with the edges
        ...         pass

        **Reporting:**

        Simple graph information is obtained using object-attributes and methods.
        Reporting typically provides views instead of containers to reduce memory
        usage. The views update as the graph is updated similarly to dict-views.
        The objects `nodes`, `edges` and `adj` provide access to data attributes
        via lookup (e.g. `nodes[n]`, `edges[u, v]`, `adj[u][v]`) and iteration
        (e.g. `nodes.items()`, `nodes.data('color')`,
        `nodes.data('color', default='blue')` and similarly for `edges`)
        Views exist for `nodes`, `edges`, `neighbors()`/`adj` and `degree`.

        For details on these and other miscellaneous methods, see below.

        **Subclasses (Advanced):**

        The Graph class uses a dict-of-dict-of-dict data structure.
        The outer dict (node_dict) holds adjacency information keyed by node.
        The next dict (adjlist_dict) represents the adjacency information and holds
        edge data keyed by neighbor.  The inner dict (edge_attr_dict) represents
        the edge data and holds edge attribute values keyed by attribute names.

        Each of these three dicts can be replaced in a subclass by a user defined
        dict-like object. In general, the dict-like features should be
        maintained but extra features can be added. To replace one of the
        dicts create a new graph class by changing the class(!) variable
        holding the factory for that dict-like structure.

        node_dict_factory : function, (default: dict)
            Factory function to be used to create the dict containing node
            attributes, keyed by node id.
            It should require no arguments and return a dict-like object

        node_attr_dict_factory: function, (default: dict)
            Factory function to be used to create the node attribute
            dict which holds attribute values keyed by attribute name.
            It should require no arguments and return a dict-like object

        adjlist_outer_dict_factory : function, (default: dict)
            Factory function to be used to create the outer-most dict
            in the data structure that holds adjacency info keyed by node.
            It should require no arguments and return a dict-like object.

        adjlist_inner_dict_factory : function, (default: dict)
            Factory function to be used to create the adjacency list
            dict which holds edge data keyed by neighbor.
            It should require no arguments and return a dict-like object

        edge_attr_dict_factory : function, (default: dict)
            Factory function to be used to create the edge attribute
            dict which holds attribute values keyed by attribute name.
            It should require no arguments and return a dict-like object.

        graph_attr_dict_factory : function, (default: dict)
            Factory function to be used to create the graph attribute
            dict which holds attribute values keyed by attribute name.
            It should require no arguments and return a dict-like object.

        Typically, if your extension doesn't impact the data structure all
        methods will inherit without issue except: `to_directed/to_undirected`.
        By default these methods create a DiGraph/Graph class and you probably
        want them to create your extension of a DiGraph/Graph. To facilitate
        this we define two class variables that you can set in your subclass.

        to_directed_class : callable, (default: DiGraph or MultiDiGraph)
            Class to create a new graph structure in the `to_directed` method.
            If `None`, a NetworkX class (DiGraph or MultiDiGraph) is used.

        to_undirected_class : callable, (default: Graph or MultiGraph)
            Class to create a new graph structure in the `to_undirected` method.
            If `None`, a NetworkX class (Graph or MultiGraph) is used.

        **Subclassing Example**

        Create a low memory graph class that effectively disallows edge
        attributes by using a single attribute dict for all edges.
        This reduces the memory used, but you lose edge attributes.

        >>> class ThinGraph(nx.Graph):
        ...     all_edge_dict = {"weight": 1}
        ...
        ...     def single_edge_dict(self):
        ...         return self.all_edge_dict
        ...
        ...     edge_attr_dict_factory = single_edge_dict
        >>> G = ThinGraph()
        >>> G.add_edge(2, 1)
        >>> G[2][1]
        {'weight': 1}
        >>> G.add_edge(2, 2)
        >>> G[2][1] is G[2][2]
        True
        """

        __networkx_backend__ = "networkx"

        _adj = _CachedPropertyResetterAdj()
        _node = _CachedPropertyResetterNode()

        node_dict_factory = dict
        node_attr_dict_factory = dict
        adjlist_outer_dict_factory = dict
        adjlist_inner_dict_factory = dict
        edge_attr_dict_factory = dict
        graph_attr_dict_factory = dict

        def to_directed_class(self):
            """Returns the class to use for empty directed copies.

            If you subclass the base classes, use this to designate
            what directed class to use for `to_directed()` copies.
            """
            return nx.DiGraph

        def to_undirected_class(self):
            """Returns the class to use for empty undirected copies.

            If you subclass the base classes, use this to designate
            what directed class to use for `to_directed()` copies.
            """
            return Graph

        def __init__(self, incoming_graph_data=None, **attr):
            """Initialize a graph with edges, name, or graph attributes.

            Parameters
            ----------
            incoming_graph_data : input graph (optional, default: None)
                Data to initialize graph. If None (default) an empty
                graph is created.  The data can be an edge list, or any
                NetworkX graph object.  If the corresponding optional Python
                packages are installed the data can also be a 2D NumPy array, a
                SciPy sparse array, or a PyGraphviz graph.

            attr : keyword arguments, optional (default= no attributes)
                Attributes to add to graph as key=value pairs.

            See Also
            --------
            convert

            Examples
            --------
            >>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G = nx.Graph(name="my graph")
            >>> e = [(1, 2), (2, 3), (3, 4)]  # list of edges
            >>> G = nx.Graph(e)

            Arbitrary graph attribute pairs (key=value) may be assigned

            >>> G = nx.Graph(e, day="Friday")
            >>> G.graph
            {'day': 'Friday'}

            """
            self.graph = self.graph_attr_dict_factory()  # dictionary for graph attributes
            self._node = self.node_dict_factory()  # empty node attribute dict
            self._adj = self.adjlist_outer_dict_factory()  # empty adjacency dict
            self.__networkx_cache__ = {}
            # attempt to load graph with data
            if incoming_graph_data is not None:
                convert.to_networkx_graph(incoming_graph_data, create_using=self)
            # load graph attributes (must be after convert)
            self.graph.update(attr)

        @cached_property
        def adj(self):
            """Graph adjacency object holding the neighbors of each node.

            This object is a read-only dict-like structure with node keys
            and neighbor-dict values.  The neighbor-dict is keyed by neighbor
            to the edge-data-dict.  So `G.adj[3][2]['color'] = 'blue'` sets
            the color of the edge `(3, 2)` to `"blue"`.

            Iterating over G.adj behaves like a dict. Useful idioms include
            `for nbr, datadict in G.adj[n].items():`.

            The neighbor information is also provided by subscripting the graph.
            So `for nbr, foovalue in G[node].data('foo', default=1):` works.

            For directed graphs, `G.adj` holds outgoing (successor) info.
            """
            return AdjacencyView(self._adj)

        @property
        def name(self):
            """String identifier of the graph.

            This graph attribute appears in the attribute dict G.graph
            keyed by the string `"name"`. as well as an attribute (technically
            a property) `G.name`. This is entirely user controlled.
            """
            return self.graph.get("name", "")

        @name.setter
        def name(self, s):
            self.graph["name"] = s
            nx._clear_cache(self)

        def __str__(self):
            """Returns a short summary of the graph.

            Returns
            -------
            info : string
                Graph information including the graph name (if any), graph type, and the
                number of nodes and edges.

            Examples
            --------
            >>> G = nx.Graph(name="foo")
            >>> str(G)
            "Graph named 'foo' with 0 nodes and 0 edges"

            >>> G = nx.path_graph(3)
            >>> str(G)
            'Graph with 3 nodes and 2 edges'

            """
            return "".join(
                [
                    type(self).__name__,
                    f" named {self.name!r}" if self.name else "",
                    f" with {self.number_of_nodes()} nodes and {self.number_of_edges()} edges",
                ]
            )

        def __iter__(self):
            """Iterate over the nodes. Use: 'for n in G'.

            Returns
            -------
            niter : iterator
                An iterator over all nodes in the graph.

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> [n for n in G]
            [0, 1, 2, 3]
            >>> list(G)
            [0, 1, 2, 3]
            """
            return iter(self._node)

        def __contains__(self, n):
            """Returns True if n is a node, False otherwise. Use: 'n in G'.

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> 1 in G
            True
            """
            try:
                return n in self._node
            except TypeError:
                return False

        def __len__(self):
            """Returns the number of nodes in the graph. Use: 'len(G)'.

            Returns
            -------
            nnodes : int
                The number of nodes in the graph.

            See Also
            --------
            number_of_nodes: identical method
            order: identical method

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> len(G)
            4

            """
            return len(self._node)

        def __getitem__(self, n):
            """Returns a dict of neighbors of node n.  Use: 'G[n]'.

            Parameters
            ----------
            n : node
               A node in the graph.

            Returns
            -------
            adj_dict : dictionary
               The adjacency dictionary for nodes connected to n.

            Notes
            -----
            G[n] is the same as G.adj[n] and similar to G.neighbors(n)
            (which is an iterator over G.adj[n])

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G[0]
            AtlasView({1: {}})
            """
            return self.adj[n]

        def add_node(self, node_for_adding, **attr):
            """Add a single node `node_for_adding` and update node attributes.

            Parameters
            ----------
            node_for_adding : node
                A node can be any hashable Python object except None.
            attr : keyword arguments, optional
                Set or change node attributes using key=value.

            See Also
            --------
            add_nodes_from

            Examples
            --------
            >>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.add_node(1)
            >>> G.add_node("Hello")
            >>> K3 = nx.Graph([(0, 1), (1, 2), (2, 0)])
            >>> G.add_node(K3)
            >>> G.number_of_nodes()
            3

            Use keywords set/change node attributes:

            >>> G.add_node(1, size=10)
            >>> G.add_node(3, weight=0.4, UTM=("13S", 382871, 3972649))

            Notes
            -----
            A hashable object is one that can be used as a key in a Python
            dictionary. This includes strings, numbers, tuples of strings
            and numbers, etc.

            On many platforms hashable items also include mutables such as
            NetworkX Graphs, though one should be careful that the hash
            doesn't change on mutables.
            """
            if node_for_adding not in self._node:
                if node_for_adding is None:
                    raise ValueError("None cannot be a node")
                self._adj[node_for_adding] = self.adjlist_inner_dict_factory()
                attr_dict = self._node[node_for_adding] = self.node_attr_dict_factory()
                attr_dict.update(attr)
            else:  # update attr even if node already exists
                self._node[node_for_adding].update(attr)
            nx._clear_cache(self)

        def add_nodes_from(self, nodes_for_adding, **attr):
            """Add multiple nodes.

            Parameters
            ----------
            nodes_for_adding : iterable container
                A container of nodes (list, dict, set, etc.).
                OR
                A container of (node, attribute dict) tuples.
                Node attributes are updated using the attribute dict.
            attr : keyword arguments, optional (default= no attributes)
                Update attributes for all nodes in nodes.
                Node attributes specified in nodes as a tuple take
                precedence over attributes specified via keyword arguments.

            See Also
            --------
            add_node

            Notes
            -----
            When adding nodes from an iterator over the graph you are changing,
            a `RuntimeError` can be raised with message:
            `RuntimeError: dictionary changed size during iteration`. This
            happens when the graph's underlying dictionary is modified during
            iteration. To avoid this error, evaluate the iterator into a separate
            object, e.g. by using `list(iterator_of_nodes)`, and pass this
            object to `G.add_nodes_from`.

            Examples
            --------
            >>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.add_nodes_from("Hello")
            >>> K3 = nx.Graph([(0, 1), (1, 2), (2, 0)])
            >>> G.add_nodes_from(K3)
            >>> sorted(G.nodes(), key=str)
            [0, 1, 2, 'H', 'e', 'l', 'o']

            Use keywords to update specific node attributes for every node.

            >>> G.add_nodes_from([1, 2], size=10)
            >>> G.add_nodes_from([3, 4], weight=0.4)

            Use (node, attrdict) tuples to update attributes for specific nodes.

            >>> G.add_nodes_from([(1, dict(size=11)), (2, {"color": "blue"})])
            >>> G.nodes[1]["size"]
            11
            >>> H = nx.Graph()
            >>> H.add_nodes_from(G.nodes(data=True))
            >>> H.nodes[1]["size"]
            11

            Evaluate an iterator over a graph if using it to modify the same graph

            >>> G = nx.Graph([(0, 1), (1, 2), (3, 4)])
            >>> # wrong way - will raise RuntimeError
            >>> # G.add_nodes_from(n + 1 for n in G.nodes)
            >>> # correct way
            >>> G.add_nodes_from(list(n + 1 for n in G.nodes))
            """
            for n in nodes_for_adding:
                try:
                    newnode = n not in self._node
                    newdict = attr
                except TypeError:
                    n, ndict = n
                    newnode = n not in self._node
                    newdict = attr.copy()
                    newdict.update(ndict)
                if newnode:
                    if n is None:
                        raise ValueError("None cannot be a node")
                    self._adj[n] = self.adjlist_inner_dict_factory()
                    self._node[n] = self.node_attr_dict_factory()
                self._node[n].update(newdict)
            nx._clear_cache(self)

        def remove_node(self, n):
            """Remove node n.

            Removes the node n and all adjacent edges.
            Attempting to remove a nonexistent node will raise an exception.

            Parameters
            ----------
            n : node
               A node in the graph

            Raises
            ------
            NetworkXError
               If n is not in the graph.

            See Also
            --------
            remove_nodes_from

            Examples
            --------
            >>> G = nx.path_graph(3)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> list(G.edges)
            [(0, 1), (1, 2)]
            >>> G.remove_node(1)
            >>> list(G.edges)
            []

            """
            adj = self._adj
            try:
                nbrs = list(adj[n])  # list handles self-loops (allows mutation)
                del self._node[n]
            except KeyError as err:  # NetworkXError if n not in self
                raise NetworkXError(f"The node {n} is not in the graph.") from err
            for u in nbrs:
                del adj[u][n]  # remove all edges n-u in graph
            del adj[n]  # now remove node
            nx._clear_cache(self)

        def remove_nodes_from(self, nodes):
            """Remove multiple nodes.

            Parameters
            ----------
            nodes : iterable container
                A container of nodes (list, dict, set, etc.).  If a node
                in the container is not in the graph it is silently
                ignored.

            See Also
            --------
            remove_node

            Notes
            -----
            When removing nodes from an iterator over the graph you are changing,
            a `RuntimeError` will be raised with message:
            `RuntimeError: dictionary changed size during iteration`. This
            happens when the graph's underlying dictionary is modified during
            iteration. To avoid this error, evaluate the iterator into a separate
            object, e.g. by using `list(iterator_of_nodes)`, and pass this
            object to `G.remove_nodes_from`.

            Examples
            --------
            >>> G = nx.path_graph(3)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> e = list(G.nodes)
            >>> e
            [0, 1, 2]
            >>> G.remove_nodes_from(e)
            >>> list(G.nodes)
            []

            Evaluate an iterator over a graph if using it to modify the same graph

            >>> G = nx.Graph([(0, 1), (1, 2), (3, 4)])
            >>> # this command will fail, as the graph's dict is modified during iteration
            >>> # G.remove_nodes_from(n for n in G.nodes if n < 2)
            >>> # this command will work, since the dictionary underlying graph is not modified
            >>> G.remove_nodes_from(list(n for n in G.nodes if n < 2))
            """
            adj = self._adj
            for n in nodes:
                try:
                    del self._node[n]
                    for u in list(adj[n]):  # list handles self-loops
                        del adj[u][n]  # (allows mutation of dict in loop)
                    del adj[n]
                except KeyError:
                    pass
            nx._clear_cache(self)

        @cached_property
        def nodes(self):
            """A NodeView of the Graph as G.nodes or G.nodes().

            Can be used as `G.nodes` for data lookup and for set-like operations.
            Can also be used as `G.nodes(data='color', default=None)` to return a
            NodeDataView which reports specific node data but no set operations.
            It presents a dict-like interface as well with `G.nodes.items()`
            iterating over `(node, nodedata)` 2-tuples and `G.nodes[3]['foo']`
            providing the value of the `foo` attribute for node `3`. In addition,
            a view `G.nodes.data('foo')` provides a dict-like interface to the
            `foo` attribute of each node. `G.nodes.data('foo', default=1)`
            provides a default for nodes that do not have attribute `foo`.

            Parameters
            ----------
            data : string or bool, optional (default=False)
                The node attribute returned in 2-tuple (n, ddict[data]).
                If True, return entire node attribute dict as (n, ddict).
                If False, return just the nodes n.

            default : value, optional (default=None)
                Value used for nodes that don't have the requested attribute.
                Only relevant if data is not True or False.

            Returns
            -------
            NodeView
                Allows set-like operations over the nodes as well as node
                attribute dict lookup and calling to get a NodeDataView.
                A NodeDataView iterates over `(n, data)` and has no set operations.
                A NodeView iterates over `n` and includes set operations.

                When called, if data is False, an iterator over nodes.
                Otherwise an iterator of 2-tuples (node, attribute value)
                where the attribute is specified in `data`.
                If data is True then the attribute becomes the
                entire data dictionary.

            Notes
            -----
            If your node data is not needed, it is simpler and equivalent
            to use the expression ``for n in G``, or ``list(G)``.

            Examples
            --------
            There are two simple ways of getting a list of all nodes in the graph:

            >>> G = nx.path_graph(3)
            >>> list(G.nodes)
            [0, 1, 2]
            >>> list(G)
            [0, 1, 2]

            To get the node data along with the nodes:

            >>> G.add_node(1, time="5pm")
            >>> G.nodes[0]["foo"] = "bar"
            >>> list(G.nodes(data=True))
            [(0, {'foo': 'bar'}), (1, {'time': '5pm'}), (2, {})]
            >>> list(G.nodes.data())
            [(0, {'foo': 'bar'}), (1, {'time': '5pm'}), (2, {})]

            >>> list(G.nodes(data="foo"))
            [(0, 'bar'), (1, None), (2, None)]
            >>> list(G.nodes.data("foo"))
            [(0, 'bar'), (1, None), (2, None)]

            >>> list(G.nodes(data="time"))
            [(0, None), (1, '5pm'), (2, None)]
            >>> list(G.nodes.data("time"))
            [(0, None), (1, '5pm'), (2, None)]

            >>> list(G.nodes(data="time", default="Not Available"))
            [(0, 'Not Available'), (1, '5pm'), (2, 'Not Available')]
            >>> list(G.nodes.data("time", default="Not Available"))
            [(0, 'Not Available'), (1, '5pm'), (2, 'Not Available')]

            If some of your nodes have an attribute and the rest are assumed
            to have a default attribute value you can create a dictionary
            from node/attribute pairs using the `default` keyword argument
            to guarantee the value is never None::

                >>> G = nx.Graph()
                >>> G.add_node(0)
                >>> G.add_node(1, weight=2)
                >>> G.add_node(2, weight=3)
                >>> dict(G.nodes(data="weight", default=1))
                {0: 1, 1: 2, 2: 3}

            """
            return NodeView(self)

        def number_of_nodes(self):
            """Returns the number of nodes in the graph.

            Returns
            -------
            nnodes : int
                The number of nodes in the graph.

            See Also
            --------
            order: identical method
            __len__: identical method

            Examples
            --------
            >>> G = nx.path_graph(3)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.number_of_nodes()
            3
            """
            return len(self._node)

        def order(self):
            """Returns the number of nodes in the graph.

            Returns
            -------
            nnodes : int
                The number of nodes in the graph.

            See Also
            --------
            number_of_nodes: identical method
            __len__: identical method

            Examples
            --------
            >>> G = nx.path_graph(3)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.order()
            3
            """
            return len(self._node)

        def has_node(self, n):
            """Returns True if the graph contains the node n.

            Identical to `n in G`

            Parameters
            ----------
            n : node

            Examples
            --------
            >>> G = nx.path_graph(3)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.has_node(0)
            True

            It is more readable and simpler to use

            >>> 0 in G
            True

            """
            try:
                return n in self._node
            except TypeError:
                return False

        def add_edge(self, u_of_edge, v_of_edge, **attr):
            """Add an edge between u and v.

            The nodes u and v will be automatically added if they are
            not already in the graph.

            Edge attributes can be specified with keywords or by directly
            accessing the edge's attribute dictionary. See examples below.

            Parameters
            ----------
            u_of_edge, v_of_edge : nodes
                Nodes can be, for example, strings or numbers.
                Nodes must be hashable (and not None) Python objects.
            attr : keyword arguments, optional
                Edge data (or labels or objects) can be assigned using
                keyword arguments.

            See Also
            --------
            add_edges_from : add a collection of edges

            Notes
            -----
            Adding an edge that already exists updates the edge data.

            Many NetworkX algorithms designed for weighted graphs use
            an edge attribute (by default `weight`) to hold a numerical value.

            Examples
            --------
            The following all add the edge e=(1, 2) to graph G:

            >>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> e = (1, 2)
            >>> G.add_edge(1, 2)  # explicit two-node form
            >>> G.add_edge(*e)  # single edge as tuple of two nodes
            >>> G.add_edges_from([(1, 2)])  # add edges from iterable container

            Associate data to edges using keywords:

            >>> G.add_edge(1, 2, weight=3)
            >>> G.add_edge(1, 3, weight=7, capacity=15, length=342.7)

            For non-string attribute keys, use subscript notation.

            >>> G.add_edge(1, 2)
            >>> G[1][2].update({0: 5})
            >>> G.edges[1, 2].update({0: 5})
            """
            u, v = u_of_edge, v_of_edge
            # add nodes
            if u not in self._node:
                if u is None:
                    raise ValueError("None cannot be a node")
                self._adj[u] = self.adjlist_inner_dict_factory()
                self._node[u] = self.node_attr_dict_factory()
            if v not in self._node:
                if v is None:
                    raise ValueError("None cannot be a node")
                self._adj[v] = self.adjlist_inner_dict_factory()
                self._node[v] = self.node_attr_dict_factory()
            # add the edge
            datadict = self._adj[u].get(v, self.edge_attr_dict_factory())
            datadict.update(attr)
            self._adj[u][v] = datadict
            self._adj[v][u] = datadict
            nx._clear_cache(self)

        def add_edges_from(self, ebunch_to_add, **attr):
            """Add all the edges in ebunch_to_add.

            Parameters
            ----------
            ebunch_to_add : container of edges
                Each edge given in the container will be added to the
                graph. The edges must be given as 2-tuples (u, v) or
                3-tuples (u, v, d) where d is a dictionary containing edge data.
            attr : keyword arguments, optional
                Edge data (or labels or objects) can be assigned using
                keyword arguments.

            See Also
            --------
            add_edge : add a single edge
            add_weighted_edges_from : convenient way to add weighted edges

            Notes
            -----
            Adding the same edge twice has no effect but any edge data
            will be updated when each duplicate edge is added.

            Edge attributes specified in an ebunch take precedence over
            attributes specified via keyword arguments.

            When adding edges from an iterator over the graph you are changing,
            a `RuntimeError` can be raised with message:
            `RuntimeError: dictionary changed size during iteration`. This
            happens when the graph's underlying dictionary is modified during
            iteration. To avoid this error, evaluate the iterator into a separate
            object, e.g. by using `list(iterator_of_edges)`, and pass this
            object to `G.add_edges_from`.

            Examples
            --------
            >>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.add_edges_from([(0, 1), (1, 2)])  # using a list of edge tuples
            >>> e = zip(range(0, 3), range(1, 4))
            >>> G.add_edges_from(e)  # Add the path graph 0-1-2-3

            Associate data to edges

            >>> G.add_edges_from([(1, 2), (2, 3)], weight=3)
            >>> G.add_edges_from([(3, 4), (1, 4)], label="WN2898")

            Evaluate an iterator over a graph if using it to modify the same graph

            >>> G = nx.Graph([(1, 2), (2, 3), (3, 4)])
            >>> # Grow graph by one new node, adding edges to all existing nodes.
            >>> # wrong way - will raise RuntimeError
            >>> # G.add_edges_from(((5, n) for n in G.nodes))
            >>> # correct way - note that there will be no self-edge for node 5
            >>> G.add_edges_from(list((5, n) for n in G.nodes))
            """
            for e in ebunch_to_add:
                ne = len(e)
                if ne == 3:
                    u, v, dd = e
                elif ne == 2:
                    u, v = e
                    dd = {}  # doesn't need edge_attr_dict_factory
                else:
                    raise NetworkXError(f"Edge tuple {e} must be a 2-tuple or 3-tuple.")
                if u not in self._node:
                    if u is None:
                        raise ValueError("None cannot be a node")
                    self._adj[u] = self.adjlist_inner_dict_factory()
                    self._node[u] = self.node_attr_dict_factory()
                if v not in self._node:
                    if v is None:
                        raise ValueError("None cannot be a node")
                    self._adj[v] = self.adjlist_inner_dict_factory()
                    self._node[v] = self.node_attr_dict_factory()
                datadict = self._adj[u].get(v, self.edge_attr_dict_factory())
                datadict.update(attr)
                datadict.update(dd)
                self._adj[u][v] = datadict
                self._adj[v][u] = datadict
            nx._clear_cache(self)

        def add_weighted_edges_from(self, ebunch_to_add, weight="weight", **attr):
            """Add weighted edges in `ebunch_to_add` with specified weight attr

            Parameters
            ----------
            ebunch_to_add : container of edges
                Each edge given in the list or container will be added
                to the graph. The edges must be given as 3-tuples (u, v, w)
                where w is a number.
            weight : string, optional (default= 'weight')
                The attribute name for the edge weights to be added.
            attr : keyword arguments, optional (default= no attributes)
                Edge attributes to add/update for all edges.

            See Also
            --------
            add_edge : add a single edge
            add_edges_from : add multiple edges

            Notes
            -----
            Adding the same edge twice for Graph/DiGraph simply updates
            the edge data. For MultiGraph/MultiDiGraph, duplicate edges
            are stored.

            When adding edges from an iterator over the graph you are changing,
            a `RuntimeError` can be raised with message:
            `RuntimeError: dictionary changed size during iteration`. This
            happens when the graph's underlying dictionary is modified during
            iteration. To avoid this error, evaluate the iterator into a separate
            object, e.g. by using `list(iterator_of_edges)`, and pass this
            object to `G.add_weighted_edges_from`.

            Examples
            --------
            >>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.add_weighted_edges_from([(0, 1, 3.0), (1, 2, 7.5)])

            Evaluate an iterator over edges before passing it

            >>> G = nx.Graph([(1, 2), (2, 3), (3, 4)])
            >>> weight = 0.1
            >>> # Grow graph by one new node, adding edges to all existing nodes.
            >>> # wrong way - will raise RuntimeError
            >>> # G.add_weighted_edges_from(((5, n, weight) for n in G.nodes))
            >>> # correct way - note that there will be no self-edge for node 5
            >>> G.add_weighted_edges_from(list((5, n, weight) for n in G.nodes))
            """
            self.add_edges_from(((u, v, {weight: d}) for u, v, d in ebunch_to_add), **attr)
            nx._clear_cache(self)

        def remove_edge(self, u, v):
            """Remove the edge between u and v.

            Parameters
            ----------
            u, v : nodes
                Remove the edge between nodes u and v.

            Raises
            ------
            NetworkXError
                If there is not an edge between u and v.

            See Also
            --------
            remove_edges_from : remove a collection of edges

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, etc
            >>> G.remove_edge(0, 1)
            >>> e = (1, 2)
            >>> G.remove_edge(*e)  # unpacks e from an edge tuple
            >>> e = (2, 3, {"weight": 7})  # an edge with attribute data
            >>> G.remove_edge(*e[:2])  # select first part of edge tuple
            """
            try:
                del self._adj[u][v]
                if u != v:  # self-loop needs only one entry removed
                    del self._adj[v][u]
            except KeyError as err:
                raise NetworkXError(f"The edge {u}-{v} is not in the graph") from err
            nx._clear_cache(self)

        def remove_edges_from(self, ebunch):
            """Remove all edges specified in ebunch.

            Parameters
            ----------
            ebunch: list or container of edge tuples
                Each edge given in the list or container will be removed
                from the graph. The edges can be:

                    - 2-tuples (u, v) edge between u and v.
                    - 3-tuples (u, v, k) where k is ignored.

            See Also
            --------
            remove_edge : remove a single edge

            Notes
            -----
            Will fail silently if an edge in ebunch is not in the graph.

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> ebunch = [(1, 2), (2, 3)]
            >>> G.remove_edges_from(ebunch)
            """
            adj = self._adj
            for e in ebunch:
                u, v = e[:2]  # ignore edge data if present
                if u in adj and v in adj[u]:
                    del adj[u][v]
                    if u != v:  # self loop needs only one entry removed
                        del adj[v][u]
            nx._clear_cache(self)

        def update(self, edges=None, nodes=None):
            """Update the graph using nodes/edges/graphs as input.

            Like dict.update, this method takes a graph as input, adding the
            graph's nodes and edges to this graph. It can also take two inputs:
            edges and nodes. Finally it can take either edges or nodes.
            To specify only nodes the keyword `nodes` must be used.

            The collections of edges and nodes are treated similarly to
            the add_edges_from/add_nodes_from methods. When iterated, they
            should yield 2-tuples (u, v) or 3-tuples (u, v, datadict).

            Parameters
            ----------
            edges : Graph object, collection of edges, or None
                The first parameter can be a graph or some edges. If it has
                attributes `nodes` and `edges`, then it is taken to be a
                Graph-like object and those attributes are used as collections
                of nodes and edges to be added to the graph.
                If the first parameter does not have those attributes, it is
                treated as a collection of edges and added to the graph.
                If the first argument is None, no edges are added.
            nodes : collection of nodes, or None
                The second parameter is treated as a collection of nodes
                to be added to the graph unless it is None.
                If `edges is None` and `nodes is None` an exception is raised.
                If the first parameter is a Graph, then `nodes` is ignored.

            Examples
            --------
            >>> G = nx.path_graph(5)
            >>> G.update(nx.complete_graph(range(4, 10)))
            >>> from itertools import combinations
            >>> edges = (
            ...     (u, v, {"power": u * v})
            ...     for u, v in combinations(range(10, 20), 2)
            ...     if u * v < 225
            ... )
            >>> nodes = [1000]  # for singleton, use a container
            >>> G.update(edges, nodes)

            Notes
            -----
            It you want to update the graph using an adjacency structure
            it is straightforward to obtain the edges/nodes from adjacency.
            The following examples provide common cases, your adjacency may
            be slightly different and require tweaks of these examples::

            >>> # dict-of-set/list/tuple
            >>> adj = {1: {2, 3}, 2: {1, 3}, 3: {1, 2}}
            >>> e = [(u, v) for u, nbrs in adj.items() for v in nbrs]
            >>> G.update(edges=e, nodes=adj)

            >>> DG = nx.DiGraph()
            >>> # dict-of-dict-of-attribute
            >>> adj = {1: {2: 1.3, 3: 0.7}, 2: {1: 1.4}, 3: {1: 0.7}}
            >>> e = [
            ...     (u, v, {"weight": d})
            ...     for u, nbrs in adj.items()
            ...     for v, d in nbrs.items()
            ... ]
            >>> DG.update(edges=e, nodes=adj)

            >>> # dict-of-dict-of-dict
            >>> adj = {1: {2: {"weight": 1.3}, 3: {"color": 0.7, "weight": 1.2}}}
            >>> e = [
            ...     (u, v, {"weight": d})
            ...     for u, nbrs in adj.items()
            ...     for v, d in nbrs.items()
            ... ]
            >>> DG.update(edges=e, nodes=adj)

            >>> # predecessor adjacency (dict-of-set)
            >>> pred = {1: {2, 3}, 2: {3}, 3: {3}}
            >>> e = [(v, u) for u, nbrs in pred.items() for v in nbrs]

            >>> # MultiGraph dict-of-dict-of-dict-of-attribute
            >>> MDG = nx.MultiDiGraph()
            >>> adj = {
            ...     1: {2: {0: {"weight": 1.3}, 1: {"weight": 1.2}}},
            ...     3: {2: {0: {"weight": 0.7}}},
            ... }
            >>> e = [
            ...     (u, v, ekey, d)
            ...     for u, nbrs in adj.items()
            ...     for v, keydict in nbrs.items()
            ...     for ekey, d in keydict.items()
            ... ]
            >>> MDG.update(edges=e)

            See Also
            --------
            add_edges_from: add multiple edges to a graph
            add_nodes_from: add multiple nodes to a graph
            """
            if edges is not None:
                if nodes is not None:
                    self.add_nodes_from(nodes)
                    self.add_edges_from(edges)
                else:
                    # check if edges is a Graph object
                    try:
                        graph_nodes = edges.nodes
                        graph_edges = edges.edges
                    except AttributeError:
                        # edge not Graph-like
                        self.add_edges_from(edges)
                    else:  # edges is Graph-like
                        self.add_nodes_from(graph_nodes.data())
                        self.add_edges_from(graph_edges.data())
                        self.graph.update(edges.graph)
            elif nodes is not None:
                self.add_nodes_from(nodes)
            else:
                raise NetworkXError("update needs nodes or edges input")

        def has_edge(self, u, v):
            """Returns True if the edge (u, v) is in the graph.

            This is the same as `v in G[u]` without KeyError exceptions.

            Parameters
            ----------
            u, v : nodes
                Nodes can be, for example, strings or numbers.
                Nodes must be hashable (and not None) Python objects.

            Returns
            -------
            edge_ind : bool
                True if edge is in the graph, False otherwise.

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.has_edge(0, 1)  # using two nodes
            True
            >>> e = (0, 1)
            >>> G.has_edge(*e)  #  e is a 2-tuple (u, v)
            True
            >>> e = (0, 1, {"weight": 7})
            >>> G.has_edge(*e[:2])  # e is a 3-tuple (u, v, data_dictionary)
            True

            The following syntax are equivalent:

            >>> G.has_edge(0, 1)
            True
            >>> 1 in G[0]  # though this gives KeyError if 0 not in G
            True

            """
            try:
                return v in self._adj[u]
            except KeyError:
                return False

        def neighbors(self, n):
            """Returns an iterator over all neighbors of node n.

            This is identical to `iter(G[n])`

            Parameters
            ----------
            n : node
               A node in the graph

            Returns
            -------
            neighbors : iterator
                An iterator over all neighbors of node n

            Raises
            ------
            NetworkXError
                If the node n is not in the graph.

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> [n for n in G.neighbors(0)]
            [1]

            Notes
            -----
            Alternate ways to access the neighbors are ``G.adj[n]`` or ``G[n]``:

            >>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.add_edge("a", "b", weight=7)
            >>> G["a"]
            AtlasView({'b': {'weight': 7}})
            >>> G = nx.path_graph(4)
            >>> [n for n in G[0]]
            [1]
            """
            try:
                return iter(self._adj[n])
            except KeyError as err:
                raise NetworkXError(f"The node {n} is not in the graph.") from err

        @cached_property
        def edges(self):
            """An EdgeView of the Graph as G.edges or G.edges().

            edges(self, nbunch=None, data=False, default=None)

            The EdgeView provides set-like operations on the edge-tuples
            as well as edge attribute lookup. When called, it also provides
            an EdgeDataView object which allows control of access to edge
            attributes (but does not provide set-like operations).
            Hence, `G.edges[u, v]['color']` provides the value of the color
            attribute for edge `(u, v)` while
            `for (u, v, c) in G.edges.data('color', default='red'):`
            iterates through all the edges yielding the color attribute
            with default `'red'` if no color attribute exists.

            Parameters
            ----------
            nbunch : single node, container, or all nodes (default= all nodes)
                The view will only report edges from these nodes.
            data : string or bool, optional (default=False)
                The edge attribute returned in 3-tuple (u, v, ddict[data]).
                If True, return edge attribute dict in 3-tuple (u, v, ddict).
                If False, return 2-tuple (u, v).
            default : value, optional (default=None)
                Value used for edges that don't have the requested attribute.
                Only relevant if data is not True or False.

            Returns
            -------
            edges : EdgeView
                A view of edge attributes, usually it iterates over (u, v)
                or (u, v, d) tuples of edges, but can also be used for
                attribute lookup as `edges[u, v]['foo']`.

            Notes
            -----
            Nodes in nbunch that are not in the graph will be (quietly) ignored.
            For directed graphs this returns the out-edges.

            Examples
            --------
            >>> G = nx.path_graph(3)  # or MultiGraph, etc
            >>> G.add_edge(2, 3, weight=5)
            >>> [e for e in G.edges]
            [(0, 1), (1, 2), (2, 3)]
            >>> G.edges.data()  # default data is {} (empty dict)
            EdgeDataView([(0, 1, {}), (1, 2, {}), (2, 3, {'weight': 5})])
            >>> G.edges.data("weight", default=1)
            EdgeDataView([(0, 1, 1), (1, 2, 1), (2, 3, 5)])
            >>> G.edges([0, 3])  # only edges from these nodes
            EdgeDataView([(0, 1), (3, 2)])
            >>> G.edges(0)  # only edges from node 0
            EdgeDataView([(0, 1)])
            """
            return EdgeView(self)

        def get_edge_data(self, u, v, default=None):
            """Returns the attribute dictionary associated with edge (u, v).

            This is identical to `G[u][v]` except the default is returned
            instead of an exception if the edge doesn't exist.

            Parameters
            ----------
            u, v : nodes
            default:  any Python object (default=None)
                Value to return if the edge (u, v) is not found.

            Returns
            -------
            edge_dict : dictionary
                The edge attribute dictionary.

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G[0][1]
            {}

            Warning: Assigning to `G[u][v]` is not permitted.
            But it is safe to assign attributes `G[u][v]['foo']`

            >>> G[0][1]["weight"] = 7
            >>> G[0][1]["weight"]
            7
            >>> G[1][0]["weight"]
            7

            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.get_edge_data(0, 1)  # default edge data is {}
            {}
            >>> e = (0, 1)
            >>> G.get_edge_data(*e)  # tuple form
            {}
            >>> G.get_edge_data("a", "b", default=0)  # edge not in graph, return 0
            0
            """
            try:
                return self._adj[u][v]
            except KeyError:
                return default

        def adjacency(self):
            """Returns an iterator over (node, adjacency dict) tuples for all nodes.

            For directed graphs, only outgoing neighbors/adjacencies are included.

            Returns
            -------
            adj_iter : iterator
               An iterator over (node, adjacency dictionary) for all nodes in
               the graph.

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> [(n, nbrdict) for n, nbrdict in G.adjacency()]
            [(0, {1: {}}), (1, {0: {}, 2: {}}), (2, {1: {}, 3: {}}), (3, {2: {}})]

            """
            return iter(self._adj.items())

        @cached_property
        def degree(self):
            """A DegreeView for the Graph as G.degree or G.degree().

            The node degree is the number of edges adjacent to the node.
            The weighted node degree is the sum of the edge weights for
            edges incident to that node.

            This object provides an iterator for (node, degree) as well as
            lookup for the degree for a single node.

            Parameters
            ----------
            nbunch : single node, container, or all nodes (default= all nodes)
                The view will only report edges incident to these nodes.

            weight : string or None, optional (default=None)
               The name of an edge attribute that holds the numerical value used
               as a weight.  If None, then each edge has weight 1.
               The degree is the sum of the edge weights adjacent to the node.

            Returns
            -------
            DegreeView or int
                If multiple nodes are requested (the default), returns a `DegreeView`
                mapping nodes to their degree.
                If a single node is requested, returns the degree of the node as an integer.

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.degree[0]  # node 0 has degree 1
            1
            >>> list(G.degree([0, 1, 2]))
            [(0, 1), (1, 2), (2, 2)]
            """
            return DegreeView(self)

        def clear(self):
            """Remove all nodes and edges from the graph.

            This also removes the name, and all graph, node, and edge attributes.

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.clear()
            >>> list(G.nodes)
            []
            >>> list(G.edges)
            []

            """
            self._adj.clear()
            self._node.clear()
            self.graph.clear()
            nx._clear_cache(self)

        def clear_edges(self):
            """Remove all edges from the graph without altering nodes.

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.clear_edges()
            >>> list(G.nodes)
            [0, 1, 2, 3]
            >>> list(G.edges)
            []
            """
            for nbr_dict in self._adj.values():
                nbr_dict.clear()
            nx._clear_cache(self)

        def is_multigraph(self):
            """Returns True if graph is a multigraph, False otherwise."""
            return False

        def is_directed(self):
            """Returns True if graph is directed, False otherwise."""
            return False

        def copy(self, as_view=False):
            """Returns a copy of the graph.

            The copy method by default returns an independent shallow copy
            of the graph and attributes. That is, if an attribute is a
            container, that container is shared by the original an the copy.
            Use Python's `copy.deepcopy` for new containers.

            If `as_view` is True then a view is returned instead of a copy.

            Notes
            -----
            All copies reproduce the graph structure, but data attributes
            may be handled in different ways. There are four types of copies
            of a graph that people might want.

            Deepcopy -- A "deepcopy" copies the graph structure as well as
            all data attributes and any objects they might contain.
            The entire graph object is new so that changes in the copy
            do not affect the original object. (see Python's copy.deepcopy)

            Data Reference (Shallow) -- For a shallow copy the graph structure
            is copied but the edge, node and graph attribute dicts are
            references to those in the original graph. This saves
            time and memory but could cause confusion if you change an attribute
            in one graph and it changes the attribute in the other.
            NetworkX does not provide this level of shallow copy.

            Independent Shallow -- This copy creates new independent attribute
            dicts and then does a shallow copy of the attributes. That is, any
            attributes that are containers are shared between the new graph
            and the original. This is exactly what `dict.copy()` provides.
            You can obtain this style copy using:

                >>> G = nx.path_graph(5)
                >>> H = G.copy()
                >>> H = G.copy(as_view=False)
                >>> H = nx.Graph(G)
                >>> H = G.__class__(G)

            Fresh Data -- For fresh data, the graph structure is copied while
            new empty data attribute dicts are created. The resulting graph
            is independent of the original and it has no edge, node or graph
            attributes. Fresh copies are not enabled. Instead use:

                >>> H = G.__class__()
                >>> H.add_nodes_from(G)
                >>> H.add_edges_from(G.edges)

            View -- Inspired by dict-views, graph-views act like read-only
            versions of the original graph, providing a copy of the original
            structure without requiring any memory for copying the information.

            See the Python copy module for more information on shallow
            and deep copies, https://docs.python.org/3/library/copy.html.

            Parameters
            ----------
            as_view : bool, optional (default=False)
                If True, the returned graph-view provides a read-only view
                of the original graph without actually copying any data.

            Returns
            -------
            G : Graph
                A copy of the graph.

            See Also
            --------
            to_directed: return a directed copy of the graph.

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> H = G.copy()

            """
            if as_view is True:
                return nx.graphviews.generic_graph_view(self)
            G = self.__class__()
            G.graph.update(self.graph)
            G.add_nodes_from((n, d.copy()) for n, d in self._node.items())
            G.add_edges_from(
                (u, v, datadict.copy())
                for u, nbrs in self._adj.items()
                for v, datadict in nbrs.items()
            )
            return G

        def to_directed(self, as_view=False):
            """Returns a directed representation of the graph.

            Returns
            -------
            G : DiGraph
                A directed graph with the same name, same nodes, and with
                each edge (u, v, data) replaced by two directed edges
                (u, v, data) and (v, u, data).

            Notes
            -----
            This returns a "deepcopy" of the edge, node, and
            graph attributes which attempts to completely copy
            all of the data and references.

            This is in contrast to the similar D=DiGraph(G) which returns a
            shallow copy of the data.

            See the Python copy module for more information on shallow
            and deep copies, https://docs.python.org/3/library/copy.html.

            Warning: If you have subclassed Graph to use dict-like objects
            in the data structure, those changes do not transfer to the
            DiGraph created by this method.

            Examples
            --------
            >>> G = nx.Graph()  # or MultiGraph, etc
            >>> G.add_edge(0, 1)
            >>> H = G.to_directed()
            >>> list(H.edges)
            [(0, 1), (1, 0)]

            If already directed, return a (deep) copy

            >>> G = nx.DiGraph()  # or MultiDiGraph, etc
            >>> G.add_edge(0, 1)
            >>> H = G.to_directed()
            >>> list(H.edges)
            [(0, 1)]
            """
            graph_class = self.to_directed_class()
            if as_view is True:
                return nx.graphviews.generic_graph_view(self, graph_class)
            # deepcopy when not a view
            G = graph_class()
            G.graph.update(deepcopy(self.graph))
            G.add_nodes_from((n, deepcopy(d)) for n, d in self._node.items())
            G.add_edges_from(
                (u, v, deepcopy(data))
                for u, nbrs in self._adj.items()
                for v, data in nbrs.items()
            )
            return G

        def to_undirected(self, as_view=False):
            """Returns an undirected copy of the graph.

            Parameters
            ----------
            as_view : bool (optional, default=False)
              If True return a view of the original undirected graph.

            Returns
            -------
            G : Graph/MultiGraph
                A deepcopy of the graph.

            See Also
            --------
            Graph, copy, add_edge, add_edges_from

            Notes
            -----
            This returns a "deepcopy" of the edge, node, and
            graph attributes which attempts to completely copy
            all of the data and references.

            This is in contrast to the similar `G = nx.DiGraph(D)` which returns a
            shallow copy of the data.

            See the Python copy module for more information on shallow
            and deep copies, https://docs.python.org/3/library/copy.html.

            Warning: If you have subclassed DiGraph to use dict-like objects
            in the data structure, those changes do not transfer to the
            Graph created by this method.

            Examples
            --------
            >>> G = nx.path_graph(2)  # or MultiGraph, etc
            >>> H = G.to_directed()
            >>> list(H.edges)
            [(0, 1), (1, 0)]
            >>> G2 = H.to_undirected()
            >>> list(G2.edges)
            [(0, 1)]
            """
            graph_class = self.to_undirected_class()
            if as_view is True:
                return nx.graphviews.generic_graph_view(self, graph_class)
            # deepcopy when not a view
            G = graph_class()
            G.graph.update(deepcopy(self.graph))
            G.add_nodes_from((n, deepcopy(d)) for n, d in self._node.items())
            G.add_edges_from(
                (u, v, deepcopy(d))
                for u, nbrs in self._adj.items()
                for v, d in nbrs.items()
            )
            return G

        def subgraph(self, nodes):
            """Returns a SubGraph view of the subgraph induced on `nodes`.

            The induced subgraph of the graph contains the nodes in `nodes`
            and the edges between those nodes.

            Parameters
            ----------
            nodes : list, iterable
                A container of nodes which will be iterated through once.

            Returns
            -------
            G : SubGraph View
                A subgraph view of the graph. The graph structure cannot be
                changed but node/edge attributes can and are shared with the
                original graph.

            Notes
            -----
            The graph, edge and node attributes are shared with the original graph.
            Changes to the graph structure is ruled out by the view, but changes
            to attributes are reflected in the original graph.

            To create a subgraph with its own copy of the edge/node attributes use:
            G.subgraph(nodes).copy()

            For an inplace reduction of a graph to a subgraph you can remove nodes:
            G.remove_nodes_from([n for n in G if n not in set(nodes)])

            Subgraph views are sometimes NOT what you want. In most cases where
            you want to do more than simply look at the induced edges, it makes
            more sense to just create the subgraph as its own graph with code like:

            ::

                # Create a subgraph SG based on a (possibly multigraph) G
                SG = G.__class__()
                SG.add_nodes_from((n, G.nodes[n]) for n in largest_wcc)
                if SG.is_multigraph():
                    SG.add_edges_from(
                        (n, nbr, key, d)
                        for n, nbrs in G.adj.items()
                        if n in largest_wcc
                        for nbr, keydict in nbrs.items()
                        if nbr in largest_wcc
                        for key, d in keydict.items()
                    )
                else:
                    SG.add_edges_from(
                        (n, nbr, d)
                        for n, nbrs in G.adj.items()
                        if n in largest_wcc
                        for nbr, d in nbrs.items()
                        if nbr in largest_wcc
                    )
                SG.graph.update(G.graph)

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> H = G.subgraph([0, 1, 2])
            >>> list(H.edges)
            [(0, 1), (1, 2)]
            """
            induced_nodes = nx.filters.show_nodes(self.nbunch_iter(nodes))
            # if already a subgraph, don't make a chain
            subgraph = nx.subgraph_view
            if hasattr(self, "_NODE_OK"):
                return subgraph(
                    self._graph, filter_node=induced_nodes, filter_edge=self._EDGE_OK
                )
            return subgraph(self, filter_node=induced_nodes)

        def edge_subgraph(self, edges):
            """Returns the subgraph induced by the specified edges.

            The induced subgraph contains each edge in `edges` and each
            node incident to any one of those edges.

            Parameters
            ----------
            edges : iterable
                An iterable of edges in this graph.

            Returns
            -------
            G : Graph
                An edge-induced subgraph of this graph with the same edge
                attributes.

            Notes
            -----
            The graph, edge, and node attributes in the returned subgraph
            view are references to the corresponding attributes in the original
            graph. The view is read-only.

            To create a full graph version of the subgraph with its own copy
            of the edge or node attributes, use::

                G.edge_subgraph(edges).copy()

            Examples
            --------
            >>> G = nx.path_graph(5)
            >>> H = G.edge_subgraph([(0, 1), (3, 4)])
            >>> list(H.nodes)
            [0, 1, 3, 4]
            >>> list(H.edges)
            [(0, 1), (3, 4)]

            """
            return nx.edge_subgraph(self, edges)

        def size(self, weight=None):
            """Returns the number of edges or total of all edge weights.

            Parameters
            ----------
            weight : string or None, optional (default=None)
                The edge attribute that holds the numerical value used
                as a weight. If None, then each edge has weight 1.

            Returns
            -------
            size : numeric
                The number of edges or
                (if weight keyword is provided) the total weight sum.

                If weight is None, returns an int. Otherwise a float
                (or more general numeric if the weights are more general).

            See Also
            --------
            number_of_edges

            Examples
            --------
            >>> G = nx.path_graph(4)  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.size()
            3

            >>> G = nx.Graph()  # or DiGraph, MultiGraph, MultiDiGraph, etc
            >>> G.add_edge("a", "b", weight=2)
            >>> G.add_edge("b", "c", weight=4)
            >>> G.size()
            2
            >>> G.size(weight="weight")
            6.0
            """
            s = sum(d for v, d in self.degree(weight=weight))
            # If `weight` is None, the sum of the degrees is guaranteed to be
            # even, so we can perform integer division and hence return an
            # integer. Otherwise, the sum of the weighted degrees is not
            # guaranteed to be an integer, so we perform "real" division.
            return s // 2 if weight is None else s / 2

        def number_of_edges(self, u=None, v=None):
            """Returns the number of edges between two nodes.

            Parameters
            ----------
            u, v : nodes, optional (default=all edges)
                If u and v are specified, return the number of edges between
                u and v. Otherwise return the total number of all edges.

            Returns
            -------
            nedges : int
                The number of edges in the graph.  If nodes `u` and `v` are
                specified return the number of edges between those nodes. If
                the graph is directed, this only returns the number of edges
                from `u` to `v`.

            See Also
            --------
            size

            Examples
            --------
            For undirected graphs, this method counts the total number of
            edges in the graph:

            >>> G = nx.path_graph(4)
            >>> G.number_of_edges()
            3

            If you specify two nodes, this counts the total number of edges
            joining the two nodes:

            >>> G.number_of_edges(0, 1)
            1

            For directed graphs, this method can count the total number of
            directed edges from `u` to `v`:

            >>> G = nx.DiGraph()
            >>> G.add_edge(0, 1)
            >>> G.add_edge(1, 0)
            >>> G.number_of_edges(0, 1)
            1

            """
            if u is None:
                return int(self.size())
            if v in self._adj[u]:
                return 1
            return 0

        def nbunch_iter(self, nbunch=None):
            """Returns an iterator over nodes contained in nbunch that are
            also in the graph.

            The nodes in nbunch are checked for membership in the graph
            and if not are silently ignored.

            Parameters
            ----------
            nbunch : single node, container, or all nodes (default= all nodes)
                The view will only report edges incident to these nodes.

            Returns
            -------
            niter : iterator
                An iterator over nodes in nbunch that are also in the graph.
                If nbunch is None, iterate over all nodes in the graph.

            Raises
            ------
            NetworkXError
                If nbunch is not a node or sequence of nodes.
                If a node in nbunch is not hashable.

            See Also
            --------
            Graph.__iter__

            Notes
            -----
            When nbunch is an iterator, the returned iterator yields values
            directly from nbunch, becoming exhausted when nbunch is exhausted.

            To test whether nbunch is a single node, one can use
            "if nbunch in self:", even after processing with this routine.

            If nbunch is not a node or a (possibly empty) sequence/iterator
            or None, a :exc:`NetworkXError` is raised.  Also, if any object in
            nbunch is not hashable, a :exc:`NetworkXError` is raised.
            """
            if nbunch is None:  # include all nodes via iterator
                bunch = iter(self._adj)
            elif nbunch in self:  # if nbunch is a single node
                bunch = iter([nbunch])
            else:  # if nbunch is a sequence of nodes

                def bunch_iter(nlist, adj):
                    try:
                        for n in nlist:
                            if n in adj:
                                yield n
                    except TypeError as err:
                        exc, message = err, err.args[0]
                        # capture error for non-sequence/iterator nbunch.
                        if "iter" in message:
                            exc = NetworkXError(
                                "nbunch is not a node or a sequence of nodes."
                            )
                        # capture error for unhashable node.
                        if "hashable" in message:
                            exc = NetworkXError(
                                f"Node {n} in sequence nbunch is not a valid node."
                            )
                        raise exc

                bunch = bunch_iter(nbunch, self._adj)
            return bunch