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# Source code for networkx.algorithms.isomorphism.vf2pp

    """
    ***************
    VF2++ Algorithm
    ***************

    An implementation of the VF2++ algorithm [1]_ for Graph Isomorphism testing.

    The simplest interface to use this module is to call:

    `vf2pp_is_isomorphic`: to check whether two graphs are isomorphic.
    `vf2pp_isomorphism`: to obtain the node mapping between two graphs,
    in case they are isomorphic.
    `vf2pp_all_isomorphisms`: to generate all possible mappings between two graphs,
    if isomorphic.

    Introduction
    ------------
    The VF2++ algorithm, follows a similar logic to that of VF2, while also
    introducing new easy-to-check cutting rules and determining the optimal access
    order of nodes. It is also implemented in a non-recursive manner, which saves
    both time and space, when compared to its previous counterpart.

    The optimal node ordering is obtained after taking into consideration both the
    degree but also the label rarity of each node.
    This way we place the nodes that are more likely to match, first in the order,
    thus examining the most promising branches in the beginning.
    The rules also consider node labels, making it easier to prune unfruitful
    branches early in the process.

    Examples
    --------

    Suppose G1 and G2 are Isomorphic Graphs. Verification is as follows:

    Without node labels:

    >>> import networkx as nx
    >>> G1 = nx.path_graph(4)
    >>> G2 = nx.path_graph(4)
    >>> nx.vf2pp_is_isomorphic(G1, G2, node_label=None)
    True
    >>> nx.vf2pp_isomorphism(G1, G2, node_label=None)
    {1: 1, 2: 2, 0: 0, 3: 3}

    With node labels:

    >>> G1 = nx.path_graph(4)
    >>> G2 = nx.path_graph(4)
    >>> mapped = {1: 1, 2: 2, 3: 3, 0: 0}
    >>> nx.set_node_attributes(
    ...     G1, dict(zip(G1, ["blue", "red", "green", "yellow"])), "label"
    ... )
    >>> nx.set_node_attributes(
    ...     G2,
    ...     dict(zip([mapped[u] for u in G1], ["blue", "red", "green", "yellow"])),
    ...     "label",
    ... )
    >>> nx.vf2pp_is_isomorphic(G1, G2, node_label="label")
    True
    >>> nx.vf2pp_isomorphism(G1, G2, node_label="label")
    {1: 1, 2: 2, 0: 0, 3: 3}

    References
    ----------
    .. [1] Jüttner, Alpár & Madarasi, Péter. (2018). "VF2++—An improved subgraph
       isomorphism algorithm". Discrete Applied Mathematics. 242.
       https://doi.org/10.1016/j.dam.2018.02.018

    """

    import collections

    import networkx as nx

    __all__ = ["vf2pp_isomorphism", "vf2pp_is_isomorphic", "vf2pp_all_isomorphisms"]

    _GraphParameters = collections.namedtuple(
        "_GraphParameters",
        [
            "G1",
            "G2",
            "G1_labels",
            "G2_labels",
            "nodes_of_G1Labels",
            "nodes_of_G2Labels",
            "G2_nodes_of_degree",
        ],
    )

    _StateParameters = collections.namedtuple(
        "_StateParameters",
        [
            "mapping",
            "reverse_mapping",
            "T1",
            "T1_in",
            "T1_tilde",
            "T1_tilde_in",
            "T2",
            "T2_in",
            "T2_tilde",
            "T2_tilde_in",
        ],
    )

    @nx._dispatchable(graphs={"G1": 0, "G2": 1}, node_attrs={"node_label": "default_label"})
    def vf2pp_isomorphism(G1, G2, node_label=None, default_label=None):
        """Return an isomorphic mapping between `G1` and `G2` if it exists.

        Parameters
        ----------
        G1, G2 : NetworkX Graph or MultiGraph instances.
            The two graphs to check for isomorphism.

        node_label : str, optional
            The name of the node attribute to be used when comparing nodes.
            The default is `None`, meaning node attributes are not considered
            in the comparison. Any node that doesn't have the `node_label`
            attribute uses `default_label` instead.

        default_label : scalar
            Default value to use when a node doesn't have an attribute
            named `node_label`. Default is `None`.

        Returns
        -------
        dict or None
            Node mapping if the two graphs are isomorphic. None otherwise.
        """
        try:
            mapping = next(vf2pp_all_isomorphisms(G1, G2, node_label, default_label))
            return mapping
        except StopIteration:
            return None

    @nx._dispatchable(graphs={"G1": 0, "G2": 1}, node_attrs={"node_label": "default_label"})
    def vf2pp_is_isomorphic(G1, G2, node_label=None, default_label=None):
        """Examines whether G1 and G2 are isomorphic.

        Parameters
        ----------
        G1, G2 : NetworkX Graph or MultiGraph instances.
            The two graphs to check for isomorphism.

        node_label : str, optional
            The name of the node attribute to be used when comparing nodes.
            The default is `None`, meaning node attributes are not considered
            in the comparison. Any node that doesn't have the `node_label`
            attribute uses `default_label` instead.

        default_label : scalar
            Default value to use when a node doesn't have an attribute
            named `node_label`. Default is `None`.

        Returns
        -------
        bool
            True if the two graphs are isomorphic, False otherwise.
        """
        if vf2pp_isomorphism(G1, G2, node_label, default_label) is not None:
            return True
        return False

    @nx._dispatchable(graphs={"G1": 0, "G2": 1}, node_attrs={"node_label": "default_label"})
    def vf2pp_all_isomorphisms(G1, G2, node_label=None, default_label=None):
        """Yields all the possible mappings between G1 and G2.

        Parameters
        ----------
        G1, G2 : NetworkX Graph or MultiGraph instances.
            The two graphs to check for isomorphism.

        node_label : str, optional
            The name of the node attribute to be used when comparing nodes.
            The default is `None`, meaning node attributes are not considered
            in the comparison. Any node that doesn't have the `node_label`
            attribute uses `default_label` instead.

        default_label : scalar
            Default value to use when a node doesn't have an attribute
            named `node_label`. Default is `None`.

        Yields
        ------
        dict
            Isomorphic mapping between the nodes in `G1` and `G2`.
        """
        if G1.number_of_nodes() == 0 or G2.number_of_nodes() == 0:
            return False

        # Create the degree dicts based on graph type
        if G1.is_directed():
            G1_degree = {
                n: (in_degree, out_degree)
                for (n, in_degree), (_, out_degree) in zip(G1.in_degree, G1.out_degree)
            }
            G2_degree = {
                n: (in_degree, out_degree)
                for (n, in_degree), (_, out_degree) in zip(G2.in_degree, G2.out_degree)
            }
        else:
            G1_degree = dict(G1.degree)
            G2_degree = dict(G2.degree)

        if not G1.is_directed():
            find_candidates = _find_candidates
            restore_Tinout = _restore_Tinout
        else:
            find_candidates = _find_candidates_Di
            restore_Tinout = _restore_Tinout_Di

        # Check that both graphs have the same number of nodes and degree sequence
        if G1.order() != G2.order():
            return False
        if sorted(G1_degree.values()) != sorted(G2_degree.values()):
            return False

        # Initialize parameters and cache necessary information about degree and labels
        graph_params, state_params = _initialize_parameters(
            G1, G2, G2_degree, node_label, default_label
        )

        # Check if G1 and G2 have the same labels, and that number of nodes per label is equal between the two graphs
        if not _precheck_label_properties(graph_params):
            return False

        # Calculate the optimal node ordering
        node_order = _matching_order(graph_params)

        # Initialize the stack
        stack = []
        candidates = iter(
            find_candidates(node_order[0], graph_params, state_params, G1_degree)
        )
        stack.append((node_order[0], candidates))

        mapping = state_params.mapping
        reverse_mapping = state_params.reverse_mapping

        # Index of the node from the order, currently being examined
        matching_node = 1

        while stack:
            current_node, candidate_nodes = stack[-1]

            try:
                candidate = next(candidate_nodes)
            except StopIteration:
                # If no remaining candidates, return to a previous state, and follow another branch
                stack.pop()
                matching_node -= 1
                if stack:
                    # Pop the previously added u-v pair, and look for a different candidate _v for u
                    popped_node1, _ = stack[-1]
                    popped_node2 = mapping[popped_node1]
                    mapping.pop(popped_node1)
                    reverse_mapping.pop(popped_node2)
                    restore_Tinout(popped_node1, popped_node2, graph_params, state_params)
                continue

            if _feasibility(current_node, candidate, graph_params, state_params):
                # Terminate if mapping is extended to its full
                if len(mapping) == G2.number_of_nodes() - 1:
                    cp_mapping = mapping.copy()
                    cp_mapping[current_node] = candidate
                    yield cp_mapping
                    continue

                # Feasibility rules pass, so extend the mapping and update the parameters
                mapping[current_node] = candidate
                reverse_mapping[candidate] = current_node
                _update_Tinout(current_node, candidate, graph_params, state_params)
                # Append the next node and its candidates to the stack
                candidates = iter(
                    find_candidates(
                        node_order[matching_node], graph_params, state_params, G1_degree
                    )
                )
                stack.append((node_order[matching_node], candidates))
                matching_node += 1

    def _precheck_label_properties(graph_params):
        G1, G2, G1_labels, G2_labels, nodes_of_G1Labels, nodes_of_G2Labels, _ = graph_params
        if any(
            label not in nodes_of_G1Labels or len(nodes_of_G1Labels[label]) != len(nodes)
            for label, nodes in nodes_of_G2Labels.items()
        ):
            return False
        return True

    def _initialize_parameters(G1, G2, G2_degree, node_label=None, default_label=-1):
        """Initializes all the necessary parameters for VF2++

        Parameters
        ----------
        G1,G2: NetworkX Graph or MultiGraph instances.
            The two graphs to check for isomorphism or monomorphism

        G1_labels,G2_labels: dict
            The label of every node in G1 and G2 respectively

        Returns
        -------
        graph_params: namedtuple
            Contains all the Graph-related parameters:

            G1,G2
            G1_labels,G2_labels: dict

        state_params: namedtuple
            Contains all the State-related parameters:

            mapping: dict
                The mapping as extended so far. Maps nodes of G1 to nodes of G2

            reverse_mapping: dict
                The reverse mapping as extended so far. Maps nodes from G2 to nodes of G1. It's basically "mapping" reversed

            T1, T2: set
                Ti contains uncovered neighbors of covered nodes from Gi, i.e. nodes that are not in the mapping, but are
                neighbors of nodes that are.

            T1_out, T2_out: set
                Ti_out contains all the nodes from Gi, that are neither in the mapping nor in Ti
        """
        G1_labels = dict(G1.nodes(data=node_label, default=default_label))
        G2_labels = dict(G2.nodes(data=node_label, default=default_label))

        graph_params = _GraphParameters(
            G1,
            G2,
            G1_labels,
            G2_labels,
            nx.utils.groups(G1_labels),
            nx.utils.groups(G2_labels),
            nx.utils.groups(G2_degree),
        )

        T1, T1_in = set(), set()
        T2, T2_in = set(), set()
        if G1.is_directed():
            T1_tilde, T1_tilde_in = (
                set(G1.nodes()),
                set(),
            )  # todo: do we need Ti_tilde_in? What nodes does it have?
            T2_tilde, T2_tilde_in = set(G2.nodes()), set()
        else:
            T1_tilde, T1_tilde_in = set(G1.nodes()), set()
            T2_tilde, T2_tilde_in = set(G2.nodes()), set()

        state_params = _StateParameters(
            {},
            {},
            T1,
            T1_in,
            T1_tilde,
            T1_tilde_in,
            T2,
            T2_in,
            T2_tilde,
            T2_tilde_in,
        )

        return graph_params, state_params

    def _matching_order(graph_params):
        """The node ordering as introduced in VF2++.

        Notes
        -----
        Taking into account the structure of the Graph and the node labeling, the nodes are placed in an order such that,
        most of the unfruitful/infeasible branches of the search space can be pruned on high levels, significantly
        decreasing the number of visited states. The premise is that, the algorithm will be able to recognize
        inconsistencies early, proceeding to go deep into the search tree only if it's needed.

        Parameters
        ----------
        graph_params: namedtuple
            Contains:

                G1,G2: NetworkX Graph or MultiGraph instances.
                    The two graphs to check for isomorphism or monomorphism.

                G1_labels,G2_labels: dict
                    The label of every node in G1 and G2 respectively.

        Returns
        -------
        node_order: list
            The ordering of the nodes.
        """
        G1, G2, G1_labels, _, _, nodes_of_G2Labels, _ = graph_params
        if not G1 and not G2:
            return {}

        if G1.is_directed():
            G1 = G1.to_undirected(as_view=True)

        V1_unordered = set(G1.nodes())
        label_rarity = {label: len(nodes) for label, nodes in nodes_of_G2Labels.items()}
        used_degrees = {node: 0 for node in G1}
        node_order = []

        while V1_unordered:
            max_rarity = min(label_rarity[G1_labels[x]] for x in V1_unordered)
            rarest_nodes = [
                n for n in V1_unordered if label_rarity[G1_labels[n]] == max_rarity
            ]
            max_node = max(rarest_nodes, key=G1.degree)

            for dlevel_nodes in nx.bfs_layers(G1, max_node):
                nodes_to_add = dlevel_nodes.copy()
                while nodes_to_add:
                    max_used_degree = max(used_degrees[n] for n in nodes_to_add)
                    max_used_degree_nodes = [
                        n for n in nodes_to_add if used_degrees[n] == max_used_degree
                    ]
                    max_degree = max(G1.degree[n] for n in max_used_degree_nodes)
                    max_degree_nodes = [
                        n for n in max_used_degree_nodes if G1.degree[n] == max_degree
                    ]
                    next_node = min(
                        max_degree_nodes, key=lambda x: label_rarity[G1_labels[x]]
                    )

                    node_order.append(next_node)
                    for node in G1.neighbors(next_node):
                        used_degrees[node] += 1

                    nodes_to_add.remove(next_node)
                    label_rarity[G1_labels[next_node]] -= 1
                    V1_unordered.discard(next_node)

        return node_order

    def _find_candidates(
        u, graph_params, state_params, G1_degree
    ):  # todo: make the 4th argument the degree of u
        """Given node u of G1, finds the candidates of u from G2.

        Parameters
        ----------
        u: Graph node
            The node from G1 for which to find the candidates from G2.

        graph_params: namedtuple
            Contains all the Graph-related parameters:

            G1,G2: NetworkX Graph or MultiGraph instances.
                The two graphs to check for isomorphism or monomorphism

            G1_labels,G2_labels: dict
                The label of every node in G1 and G2 respectively

        state_params: namedtuple
            Contains all the State-related parameters:

            mapping: dict
                The mapping as extended so far. Maps nodes of G1 to nodes of G2

            reverse_mapping: dict
                The reverse mapping as extended so far. Maps nodes from G2 to nodes of G1. It's basically "mapping" reversed

            T1, T2: set
                Ti contains uncovered neighbors of covered nodes from Gi, i.e. nodes that are not in the mapping, but are
                neighbors of nodes that are.

            T1_tilde, T2_tilde: set
                Ti_tilde contains all the nodes from Gi, that are neither in the mapping nor in Ti

        Returns
        -------
        candidates: set
            The nodes from G2 which are candidates for u.
        """
        G1, G2, G1_labels, _, _, nodes_of_G2Labels, G2_nodes_of_degree = graph_params
        mapping, reverse_mapping, _, _, _, _, _, _, T2_tilde, _ = state_params

        covered_nbrs = [nbr for nbr in G1[u] if nbr in mapping]
        if not covered_nbrs:
            candidates = set(nodes_of_G2Labels[G1_labels[u]])
            candidates.intersection_update(G2_nodes_of_degree[G1_degree[u]])
            candidates.intersection_update(T2_tilde)
            candidates.difference_update(reverse_mapping)
            if G1.is_multigraph():
                candidates.difference_update(
                    {
                        node
                        for node in candidates
                        if G1.number_of_edges(u, u) != G2.number_of_edges(node, node)
                    }
                )
            return candidates

        nbr1 = covered_nbrs[0]
        common_nodes = set(G2[mapping[nbr1]])

        for nbr1 in covered_nbrs[1:]:
            common_nodes.intersection_update(G2[mapping[nbr1]])

        common_nodes.difference_update(reverse_mapping)
        common_nodes.intersection_update(G2_nodes_of_degree[G1_degree[u]])
        common_nodes.intersection_update(nodes_of_G2Labels[G1_labels[u]])
        if G1.is_multigraph():
            common_nodes.difference_update(
                {
                    node
                    for node in common_nodes
                    if G1.number_of_edges(u, u) != G2.number_of_edges(node, node)
                }
            )
        return common_nodes

    def _find_candidates_Di(u, graph_params, state_params, G1_degree):
        G1, G2, G1_labels, _, _, nodes_of_G2Labels, G2_nodes_of_degree = graph_params
        mapping, reverse_mapping, _, _, _, _, _, _, T2_tilde, _ = state_params

        covered_successors = [succ for succ in G1[u] if succ in mapping]
        covered_predecessors = [pred for pred in G1.pred[u] if pred in mapping]

        if not (covered_successors or covered_predecessors):
            candidates = set(nodes_of_G2Labels[G1_labels[u]])
            candidates.intersection_update(G2_nodes_of_degree[G1_degree[u]])
            candidates.intersection_update(T2_tilde)
            candidates.difference_update(reverse_mapping)
            if G1.is_multigraph():
                candidates.difference_update(
                    {
                        node
                        for node in candidates
                        if G1.number_of_edges(u, u) != G2.number_of_edges(node, node)
                    }
                )
            return candidates

        if covered_successors:
            succ1 = covered_successors[0]
            common_nodes = set(G2.pred[mapping[succ1]])

            for succ1 in covered_successors[1:]:
                common_nodes.intersection_update(G2.pred[mapping[succ1]])
        else:
            pred1 = covered_predecessors.pop()
            common_nodes = set(G2[mapping[pred1]])

        for pred1 in covered_predecessors:
            common_nodes.intersection_update(G2[mapping[pred1]])

        common_nodes.difference_update(reverse_mapping)
        common_nodes.intersection_update(G2_nodes_of_degree[G1_degree[u]])
        common_nodes.intersection_update(nodes_of_G2Labels[G1_labels[u]])
        if G1.is_multigraph():
            common_nodes.difference_update(
                {
                    node
                    for node in common_nodes
                    if G1.number_of_edges(u, u) != G2.number_of_edges(node, node)
                }
            )
        return common_nodes

    def _feasibility(node1, node2, graph_params, state_params):
        """Given a candidate pair of nodes u and v from G1 and G2 respectively, checks if it's feasible to extend the
        mapping, i.e. if u and v can be matched.

        Notes
        -----
        This function performs all the necessary checking by applying both consistency and cutting rules.

        Parameters
        ----------
        node1, node2: Graph node
            The candidate pair of nodes being checked for matching

        graph_params: namedtuple
            Contains all the Graph-related parameters:

            G1,G2: NetworkX Graph or MultiGraph instances.
                The two graphs to check for isomorphism or monomorphism

            G1_labels,G2_labels: dict
                The label of every node in G1 and G2 respectively

        state_params: namedtuple
            Contains all the State-related parameters:

            mapping: dict
                The mapping as extended so far. Maps nodes of G1 to nodes of G2

            reverse_mapping: dict
                The reverse mapping as extended so far. Maps nodes from G2 to nodes of G1. It's basically "mapping" reversed

            T1, T2: set
                Ti contains uncovered neighbors of covered nodes from Gi, i.e. nodes that are not in the mapping, but are
                neighbors of nodes that are.

            T1_out, T2_out: set
                Ti_out contains all the nodes from Gi, that are neither in the mapping nor in Ti

        Returns
        -------
        True if all checks are successful, False otherwise.
        """
        G1 = graph_params.G1

        if _cut_PT(node1, node2, graph_params, state_params):
            return False

        if G1.is_multigraph():
            if not _consistent_PT(node1, node2, graph_params, state_params):
                return False

        return True

    def _cut_PT(u, v, graph_params, state_params):
        """Implements the cutting rules for the ISO problem.

        Parameters
        ----------
        u, v: Graph node
            The two candidate nodes being examined.

        graph_params: namedtuple
            Contains all the Graph-related parameters:

            G1,G2: NetworkX Graph or MultiGraph instances.
                The two graphs to check for isomorphism or monomorphism

            G1_labels,G2_labels: dict
                The label of every node in G1 and G2 respectively

        state_params: namedtuple
            Contains all the State-related parameters:

            mapping: dict
                The mapping as extended so far. Maps nodes of G1 to nodes of G2

            reverse_mapping: dict
                The reverse mapping as extended so far. Maps nodes from G2 to nodes of G1. It's basically "mapping" reversed

            T1, T2: set
                Ti contains uncovered neighbors of covered nodes from Gi, i.e. nodes that are not in the mapping, but are
                neighbors of nodes that are.

            T1_tilde, T2_tilde: set
                Ti_out contains all the nodes from Gi, that are neither in the mapping nor in Ti

        Returns
        -------
        True if we should prune this branch, i.e. the node pair failed the cutting checks. False otherwise.
        """
        G1, G2, G1_labels, G2_labels, _, _, _ = graph_params
        (
            _,
            _,
            T1,
            T1_in,
            T1_tilde,
            _,
            T2,
            T2_in,
            T2_tilde,
            _,
        ) = state_params

        u_labels_predecessors, v_labels_predecessors = {}, {}
        if G1.is_directed():
            u_labels_predecessors = nx.utils.groups(
                {n1: G1_labels[n1] for n1 in G1.pred[u]}
            )
            v_labels_predecessors = nx.utils.groups(
                {n2: G2_labels[n2] for n2 in G2.pred[v]}
            )

            if set(u_labels_predecessors.keys()) != set(v_labels_predecessors.keys()):
                return True

        u_labels_successors = nx.utils.groups({n1: G1_labels[n1] for n1 in G1[u]})
        v_labels_successors = nx.utils.groups({n2: G2_labels[n2] for n2 in G2[v]})

        # if the neighbors of u, do not have the same labels as those of v, NOT feasible.
        if set(u_labels_successors.keys()) != set(v_labels_successors.keys()):
            return True

        for label, G1_nbh in u_labels_successors.items():
            G2_nbh = v_labels_successors[label]

            if G1.is_multigraph():
                # Check for every neighbor in the neighborhood, if u-nbr1 has same edges as v-nbr2
                u_nbrs_edges = sorted(G1.number_of_edges(u, x) for x in G1_nbh)
                v_nbrs_edges = sorted(G2.number_of_edges(v, x) for x in G2_nbh)
                if any(
                    u_nbr_edges != v_nbr_edges
                    for u_nbr_edges, v_nbr_edges in zip(u_nbrs_edges, v_nbrs_edges)
                ):
                    return True

            if len(T1.intersection(G1_nbh)) != len(T2.intersection(G2_nbh)):
                return True
            if len(T1_tilde.intersection(G1_nbh)) != len(T2_tilde.intersection(G2_nbh)):
                return True
            if G1.is_directed() and len(T1_in.intersection(G1_nbh)) != len(
                T2_in.intersection(G2_nbh)
            ):
                return True

        if not G1.is_directed():
            return False

        for label, G1_pred in u_labels_predecessors.items():
            G2_pred = v_labels_predecessors[label]

            if G1.is_multigraph():
                # Check for every neighbor in the neighborhood, if u-nbr1 has same edges as v-nbr2
                u_pred_edges = sorted(G1.number_of_edges(u, x) for x in G1_pred)
                v_pred_edges = sorted(G2.number_of_edges(v, x) for x in G2_pred)
                if any(
                    u_nbr_edges != v_nbr_edges
                    for u_nbr_edges, v_nbr_edges in zip(u_pred_edges, v_pred_edges)
                ):
                    return True

            if len(T1.intersection(G1_pred)) != len(T2.intersection(G2_pred)):
                return True
            if len(T1_tilde.intersection(G1_pred)) != len(T2_tilde.intersection(G2_pred)):
                return True
            if len(T1_in.intersection(G1_pred)) != len(T2_in.intersection(G2_pred)):
                return True

        return False

    def _consistent_PT(u, v, graph_params, state_params):
        """Checks the consistency of extending the mapping using the current node pair.

        Parameters
        ----------
        u, v: Graph node
            The two candidate nodes being examined.

        graph_params: namedtuple
            Contains all the Graph-related parameters:

            G1,G2: NetworkX Graph or MultiGraph instances.
                The two graphs to check for isomorphism or monomorphism

            G1_labels,G2_labels: dict
                The label of every node in G1 and G2 respectively

        state_params: namedtuple
            Contains all the State-related parameters:

            mapping: dict
                The mapping as extended so far. Maps nodes of G1 to nodes of G2

            reverse_mapping: dict
                The reverse mapping as extended so far. Maps nodes from G2 to nodes of G1. It's basically "mapping" reversed

            T1, T2: set
                Ti contains uncovered neighbors of covered nodes from Gi, i.e. nodes that are not in the mapping, but are
                neighbors of nodes that are.

            T1_out, T2_out: set
                Ti_out contains all the nodes from Gi, that are neither in the mapping nor in Ti

        Returns
        -------
        True if the pair passes all the consistency checks successfully. False otherwise.
        """
        G1, G2 = graph_params.G1, graph_params.G2
        mapping, reverse_mapping = state_params.mapping, state_params.reverse_mapping

        for neighbor in G1[u]:
            if neighbor in mapping:
                if G1.number_of_edges(u, neighbor) != G2.number_of_edges(
                    v, mapping[neighbor]
                ):
                    return False

        for neighbor in G2[v]:
            if neighbor in reverse_mapping:
                if G1.number_of_edges(u, reverse_mapping[neighbor]) != G2.number_of_edges(
                    v, neighbor
                ):
                    return False

        if not G1.is_directed():
            return True

        for predecessor in G1.pred[u]:
            if predecessor in mapping:
                if G1.number_of_edges(predecessor, u) != G2.number_of_edges(
                    mapping[predecessor], v
                ):
                    return False

        for predecessor in G2.pred[v]:
            if predecessor in reverse_mapping:
                if G1.number_of_edges(
                    reverse_mapping[predecessor], u
                ) != G2.number_of_edges(predecessor, v):
                    return False

        return True

    def _update_Tinout(new_node1, new_node2, graph_params, state_params):
        """Updates the Ti/Ti_out (i=1,2) when a new node pair u-v is added to the mapping.

        Notes
        -----
        This function should be called right after the feasibility checks are passed, and node1 is mapped to node2. The
        purpose of this function is to avoid brute force computing of Ti/Ti_out by iterating over all nodes of the graph
        and checking which nodes satisfy the necessary conditions. Instead, in every step of the algorithm we focus
        exclusively on the two nodes that are being added to the mapping, incrementally updating Ti/Ti_out.

        Parameters
        ----------
        new_node1, new_node2: Graph node
            The two new nodes, added to the mapping.

        graph_params: namedtuple
            Contains all the Graph-related parameters:

            G1,G2: NetworkX Graph or MultiGraph instances.
                The two graphs to check for isomorphism or monomorphism

            G1_labels,G2_labels: dict
                The label of every node in G1 and G2 respectively

        state_params: namedtuple
            Contains all the State-related parameters:

            mapping: dict
                The mapping as extended so far. Maps nodes of G1 to nodes of G2

            reverse_mapping: dict
                The reverse mapping as extended so far. Maps nodes from G2 to nodes of G1. It's basically "mapping" reversed

            T1, T2: set
                Ti contains uncovered neighbors of covered nodes from Gi, i.e. nodes that are not in the mapping, but are
                neighbors of nodes that are.

            T1_tilde, T2_tilde: set
                Ti_out contains all the nodes from Gi, that are neither in the mapping nor in Ti
        """
        G1, G2, _, _, _, _, _ = graph_params
        (
            mapping,
            reverse_mapping,
            T1,
            T1_in,
            T1_tilde,
            T1_tilde_in,
            T2,
            T2_in,
            T2_tilde,
            T2_tilde_in,
        ) = state_params

        uncovered_successors_G1 = {succ for succ in G1[new_node1] if succ not in mapping}
        uncovered_successors_G2 = {
            succ for succ in G2[new_node2] if succ not in reverse_mapping
        }

        # Add the uncovered neighbors of node1 and node2 in T1 and T2 respectively
        T1.update(uncovered_successors_G1)
        T2.update(uncovered_successors_G2)
        T1.discard(new_node1)
        T2.discard(new_node2)

        T1_tilde.difference_update(uncovered_successors_G1)
        T2_tilde.difference_update(uncovered_successors_G2)
        T1_tilde.discard(new_node1)
        T2_tilde.discard(new_node2)

        if not G1.is_directed():
            return

        uncovered_predecessors_G1 = {
            pred for pred in G1.pred[new_node1] if pred not in mapping
        }
        uncovered_predecessors_G2 = {
            pred for pred in G2.pred[new_node2] if pred not in reverse_mapping
        }

        T1_in.update(uncovered_predecessors_G1)
        T2_in.update(uncovered_predecessors_G2)
        T1_in.discard(new_node1)
        T2_in.discard(new_node2)

        T1_tilde.difference_update(uncovered_predecessors_G1)
        T2_tilde.difference_update(uncovered_predecessors_G2)
        T1_tilde.discard(new_node1)
        T2_tilde.discard(new_node2)

    def _restore_Tinout(popped_node1, popped_node2, graph_params, state_params):
        """Restores the previous version of Ti/Ti_out when a node pair is deleted from the mapping.

        Parameters
        ----------
        popped_node1, popped_node2: Graph node
            The two nodes deleted from the mapping.

        graph_params: namedtuple
            Contains all the Graph-related parameters:

            G1,G2: NetworkX Graph or MultiGraph instances.
                The two graphs to check for isomorphism or monomorphism

            G1_labels,G2_labels: dict
                The label of every node in G1 and G2 respectively

        state_params: namedtuple
            Contains all the State-related parameters:

            mapping: dict
                The mapping as extended so far. Maps nodes of G1 to nodes of G2

            reverse_mapping: dict
                The reverse mapping as extended so far. Maps nodes from G2 to nodes of G1. It's basically "mapping" reversed

            T1, T2: set
                Ti contains uncovered neighbors of covered nodes from Gi, i.e. nodes that are not in the mapping, but are
                neighbors of nodes that are.

            T1_tilde, T2_tilde: set
                Ti_out contains all the nodes from Gi, that are neither in the mapping nor in Ti
        """
        # If the node we want to remove from the mapping, has at least one covered neighbor, add it to T1.
        G1, G2, _, _, _, _, _ = graph_params
        (
            mapping,
            reverse_mapping,
            T1,
            T1_in,
            T1_tilde,
            T1_tilde_in,
            T2,
            T2_in,
            T2_tilde,
            T2_tilde_in,
        ) = state_params

        is_added = False
        for neighbor in G1[popped_node1]:
            if neighbor in mapping:
                # if a neighbor of the excluded node1 is in the mapping, keep node1 in T1
                is_added = True
                T1.add(popped_node1)
            else:
                # check if its neighbor has another connection with a covered node. If not, only then exclude it from T1
                if any(nbr in mapping for nbr in G1[neighbor]):
                    continue
                T1.discard(neighbor)
                T1_tilde.add(neighbor)

        # Case where the node is not present in neither the mapping nor T1. By definition, it should belong to T1_tilde
        if not is_added:
            T1_tilde.add(popped_node1)

        is_added = False
        for neighbor in G2[popped_node2]:
            if neighbor in reverse_mapping:
                is_added = True
                T2.add(popped_node2)
            else:
                if any(nbr in reverse_mapping for nbr in G2[neighbor]):
                    continue
                T2.discard(neighbor)
                T2_tilde.add(neighbor)

        if not is_added:
            T2_tilde.add(popped_node2)

    def _restore_Tinout_Di(popped_node1, popped_node2, graph_params, state_params):
        # If the node we want to remove from the mapping, has at least one covered neighbor, add it to T1.
        G1, G2, _, _, _, _, _ = graph_params
        (
            mapping,
            reverse_mapping,
            T1,
            T1_in,
            T1_tilde,
            T1_tilde_in,
            T2,
            T2_in,
            T2_tilde,
            T2_tilde_in,
        ) = state_params

        is_added = False
        for successor in G1[popped_node1]:
            if successor in mapping:
                # if a neighbor of the excluded node1 is in the mapping, keep node1 in T1
                is_added = True
                T1_in.add(popped_node1)
            else:
                # check if its neighbor has another connection with a covered node. If not, only then exclude it from T1
                if not any(pred in mapping for pred in G1.pred[successor]):
                    T1.discard(successor)

                if not any(succ in mapping for succ in G1[successor]):
                    T1_in.discard(successor)

                if successor not in T1:
                    if successor not in T1_in:
                        T1_tilde.add(successor)

        for predecessor in G1.pred[popped_node1]:
            if predecessor in mapping:
                # if a neighbor of the excluded node1 is in the mapping, keep node1 in T1
                is_added = True
                T1.add(popped_node1)
            else:
                # check if its neighbor has another connection with a covered node. If not, only then exclude it from T1
                if not any(pred in mapping for pred in G1.pred[predecessor]):
                    T1.discard(predecessor)

                if not any(succ in mapping for succ in G1[predecessor]):
                    T1_in.discard(predecessor)

                if not (predecessor in T1 or predecessor in T1_in):
                    T1_tilde.add(predecessor)

        # Case where the node is not present in neither the mapping nor T1. By definition it should belong to T1_tilde
        if not is_added:
            T1_tilde.add(popped_node1)

        is_added = False
        for successor in G2[popped_node2]:
            if successor in reverse_mapping:
                is_added = True
                T2_in.add(popped_node2)
            else:
                if not any(pred in reverse_mapping for pred in G2.pred[successor]):
                    T2.discard(successor)

                if not any(succ in reverse_mapping for succ in G2[successor]):
                    T2_in.discard(successor)

                if successor not in T2:
                    if successor not in T2_in:
                        T2_tilde.add(successor)

        for predecessor in G2.pred[popped_node2]:
            if predecessor in reverse_mapping:
                # if a neighbor of the excluded node1 is in the mapping, keep node1 in T1
                is_added = True
                T2.add(popped_node2)
            else:
                # check if its neighbor has another connection with a covered node. If not, only then exclude it from T1
                if not any(pred in reverse_mapping for pred in G2.pred[predecessor]):
                    T2.discard(predecessor)

                if not any(succ in reverse_mapping for succ in G2[predecessor]):
                    T2_in.discard(predecessor)

                if not (predecessor in T2 or predecessor in T2_in):
                    T2_tilde.add(predecessor)

        if not is_added:
            T2_tilde.add(popped_node2)