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# Source code for networkx.algorithms.asteroidal

    """
    Algorithms for asteroidal triples and asteroidal numbers in graphs.

    An asteroidal triple in a graph G is a set of three non-adjacent vertices
    u, v and w such that there exist a path between any two of them that avoids
    closed neighborhood of the third. More formally, v_j, v_k belongs to the same
    connected component of G - N[v_i], where N[v_i] denotes the closed neighborhood
    of v_i. A graph which does not contain any asteroidal triples is called
    an AT-free graph. The class of AT-free graphs is a graph class for which
    many NP-complete problems are solvable in polynomial time. Amongst them,
    independent set and coloring.
    """

    import networkx as nx
    from networkx.utils import not_implemented_for

    __all__ = ["is_at_free", "find_asteroidal_triple"]

    @not_implemented_for("directed")
    @not_implemented_for("multigraph")
    @nx._dispatchable
    def find_asteroidal_triple(G):
        r"""Find an asteroidal triple in the given graph.

        An asteroidal triple is a triple of non-adjacent vertices such that
        there exists a path between any two of them which avoids the closed
        neighborhood of the third. It checks all independent triples of vertices
        and whether they are an asteroidal triple or not. This is done with the
        help of a data structure called a component structure.
        A component structure encodes information about which vertices belongs to
        the same connected component when the closed neighborhood of a given vertex
        is removed from the graph. The algorithm used to check is the trivial
        one, outlined in [1]_, which has a runtime of
        :math:`O(|V||\overline{E} + |V||E|)`, where the second term is the
        creation of the component structure.

        Parameters
        ----------
        G : NetworkX Graph
            The graph to check whether is AT-free or not

        Returns
        -------
        list or None
            An asteroidal triple is returned as a list of nodes. If no asteroidal
            triple exists, i.e. the graph is AT-free, then None is returned.
            The returned value depends on the certificate parameter. The default
            option is a bool which is True if the graph is AT-free, i.e. the
            given graph contains no asteroidal triples, and False otherwise, i.e.
            if the graph contains at least one asteroidal triple.

        Notes
        -----
        The component structure and the algorithm is described in [1]_. The current
        implementation implements the trivial algorithm for simple graphs.

        References
        ----------
        .. [1] Ekkehard Köhler,
           "Recognizing Graphs without asteroidal triples",
           Journal of Discrete Algorithms 2, pages 439-452, 2004.
           https://www.sciencedirect.com/science/article/pii/S157086670400019X
        """
        V = set(G.nodes)

        if len(V) < 6:
            # An asteroidal triple cannot exist in a graph with 5 or less vertices.
            return None

        component_structure = create_component_structure(G)
        E_complement = set(nx.complement(G).edges)

        for e in E_complement:
            u = e[0]
            v = e[1]
            u_neighborhood = set(G[u]).union([u])
            v_neighborhood = set(G[v]).union([v])
            union_of_neighborhoods = u_neighborhood.union(v_neighborhood)
            for w in V - union_of_neighborhoods:
                # Check for each pair of vertices whether they belong to the
                # same connected component when the closed neighborhood of the
                # third is removed.
                if (
                    component_structure[u][v] == component_structure[u][w]
                    and component_structure[v][u] == component_structure[v][w]
                    and component_structure[w][u] == component_structure[w][v]
                ):
                    return [u, v, w]
        return None

    @not_implemented_for("directed")
    @not_implemented_for("multigraph")
    @nx._dispatchable
    def is_at_free(G):
        """Check if a graph is AT-free.

        The method uses the `find_asteroidal_triple` method to recognize
        an AT-free graph. If no asteroidal triple is found the graph is
        AT-free and True is returned. If at least one asteroidal triple is
        found the graph is not AT-free and False is returned.

        Parameters
        ----------
        G : NetworkX Graph
            The graph to check whether is AT-free or not.

        Returns
        -------
        bool
            True if G is AT-free and False otherwise.

        Examples
        --------
        >>> G = nx.Graph([(0, 1), (0, 2), (1, 2), (1, 3), (1, 4), (4, 5)])
        >>> nx.is_at_free(G)
        True

        >>> G = nx.cycle_graph(6)
        >>> nx.is_at_free(G)
        False
        """
        return find_asteroidal_triple(G) is None

    @not_implemented_for("directed")
    @not_implemented_for("multigraph")
    @nx._dispatchable
    def create_component_structure(G):
        r"""Create component structure for G.

        A *component structure* is an `nxn` array, denoted `c`, where `n` is
        the number of vertices,  where each row and column corresponds to a vertex.

        .. math::
            c_{uv} = \begin{cases} 0, if v \in N[u] \\
                k, if v \in component k of G \setminus N[u] \end{cases}

        Where `k` is an arbitrary label for each component. The structure is used
        to simplify the detection of asteroidal triples.

        Parameters
        ----------
        G : NetworkX Graph
            Undirected, simple graph.

        Returns
        -------
        component_structure : dictionary
            A dictionary of dictionaries, keyed by pairs of vertices.

        """
        V = set(G.nodes)
        component_structure = {}
        for v in V:
            label = 0
            closed_neighborhood = set(G[v]).union({v})
            row_dict = {}
            for u in closed_neighborhood:
                row_dict[u] = 0

            G_reduced = G.subgraph(set(G.nodes) - closed_neighborhood)
            for cc in nx.connected_components(G_reduced):
                label += 1
                for u in cc:
                    row_dict[u] = label

            component_structure[v] = row_dict

        return component_structure