venv/lib/python3.10/site-packages/networkx/generators/line.py

"""Functions for generating line graphs."""

from collections import defaultdict
from functools import partial
from itertools import combinations

import networkx as nx
from networkx.utils import arbitrary_element
from networkx.utils.decorators import not_implemented_for

__all__ = ["line_graph", "inverse_line_graph"]


@nx._dispatchable(returns_graph=True)
def line_graph(G, create_using=None):
    r"""Returns the line graph of the graph or digraph `G`.

    The line graph of a graph `G` has a node for each edge in `G` and an
    edge joining those nodes if the two edges in `G` share a common node. For
    directed graphs, nodes are adjacent exactly when the edges they represent
    form a directed path of length two.

    The nodes of the line graph are 2-tuples of nodes in the original graph (or
    3-tuples for multigraphs, with the key of the edge as the third element).

    For information about self-loops and more discussion, see the **Notes**
    section below.

    Parameters
    ----------
    G : graph
        A NetworkX Graph, DiGraph, MultiGraph, or MultiDigraph.
    create_using : NetworkX graph constructor, optional (default=nx.Graph)
       Graph type to create. If graph instance, then cleared before populated.

    Returns
    -------
    L : graph
        The line graph of G.

    Examples
    --------
    >>> G = nx.star_graph(3)
    >>> L = nx.line_graph(G)
    >>> print(sorted(map(sorted, L.edges())))  # makes a 3-clique, K3
    [[(0, 1), (0, 2)], [(0, 1), (0, 3)], [(0, 2), (0, 3)]]

    Edge attributes from `G` are not copied over as node attributes in `L`, but
    attributes can be copied manually:

    >>> G = nx.path_graph(4)
    >>> G.add_edges_from((u, v, {"tot": u + v}) for u, v in G.edges)
    >>> G.edges(data=True)
    EdgeDataView([(0, 1, {'tot': 1}), (1, 2, {'tot': 3}), (2, 3, {'tot': 5})])
    >>> H = nx.line_graph(G)
    >>> H.add_nodes_from((node, G.edges[node]) for node in H)
    >>> H.nodes(data=True)
    NodeDataView({(0, 1): {'tot': 1}, (2, 3): {'tot': 5}, (1, 2): {'tot': 3}})

    Notes
    -----
    Graph, node, and edge data are not propagated to the new graph. For
    undirected graphs, the nodes in G must be sortable, otherwise the
    constructed line graph may not be correct.

    *Self-loops in undirected graphs*

    For an undirected graph `G` without multiple edges, each edge can be
    written as a set `\{u, v\}`.  Its line graph `L` has the edges of `G` as
    its nodes. If `x` and `y` are two nodes in `L`, then `\{x, y\}` is an edge
    in `L` if and only if the intersection of `x` and `y` is nonempty. Thus,
    the set of all edges is determined by the set of all pairwise intersections
    of edges in `G`.

    Trivially, every edge in G would have a nonzero intersection with itself,
    and so every node in `L` should have a self-loop. This is not so
    interesting, and the original context of line graphs was with simple
    graphs, which had no self-loops or multiple edges. The line graph was also
    meant to be a simple graph and thus, self-loops in `L` are not part of the
    standard definition of a line graph. In a pairwise intersection matrix,
    this is analogous to excluding the diagonal entries from the line graph
    definition.

    Self-loops and multiple edges in `G` add nodes to `L` in a natural way, and
    do not require any fundamental changes to the definition. It might be
    argued that the self-loops we excluded before should now be included.
    However, the self-loops are still "trivial" in some sense and thus, are
    usually excluded.

    *Self-loops in directed graphs*

    For a directed graph `G` without multiple edges, each edge can be written
    as a tuple `(u, v)`. Its line graph `L` has the edges of `G` as its
    nodes. If `x` and `y` are two nodes in `L`, then `(x, y)` is an edge in `L`
    if and only if the tail of `x` matches the head of `y`, for example, if `x
    = (a, b)` and `y = (b, c)` for some vertices `a`, `b`, and `c` in `G`.

    Due to the directed nature of the edges, it is no longer the case that
    every edge in `G` should have a self-loop in `L`. Now, the only time
    self-loops arise is if a node in `G` itself has a self-loop.  So such
    self-loops are no longer "trivial" but instead, represent essential
    features of the topology of `G`. For this reason, the historical
    development of line digraphs is such that self-loops are included. When the
    graph `G` has multiple edges, once again only superficial changes are
    required to the definition.

    References
    ----------
    * Harary, Frank, and Norman, Robert Z., "Some properties of line digraphs",
      Rend. Circ. Mat. Palermo, II. Ser. 9 (1960), 161--168.
    * Hemminger, R. L.; Beineke, L. W. (1978), "Line graphs and line digraphs",
      in Beineke, L. W.; Wilson, R. J., Selected Topics in Graph Theory,
      Academic Press Inc., pp. 271--305.

    """
    if G.is_directed():
        L = _lg_directed(G, create_using=create_using)
    else:
        L = _lg_undirected(G, selfloops=False, create_using=create_using)
    return L


def _lg_directed(G, create_using=None):
    """Returns the line graph L of the (multi)digraph G.

    Edges in G appear as nodes in L, represented as tuples of the form (u,v)
    or (u,v,key) if G is a multidigraph. A node in L corresponding to the edge
    (u,v) is connected to every node corresponding to an edge (v,w).

    Parameters
    ----------
    G : digraph
        A directed graph or directed multigraph.
    create_using : NetworkX graph constructor, optional
       Graph type to create. If graph instance, then cleared before populated.
       Default is to use the same graph class as `G`.

    """
    L = nx.empty_graph(0, create_using, default=G.__class__)

    # Create a graph specific edge function.
    get_edges = partial(G.edges, keys=True) if G.is_multigraph() else G.edges

    for from_node in get_edges():
        # from_node is: (u,v) or (u,v,key)
        L.add_node(from_node)
        for to_node in get_edges(from_node[1]):
            L.add_edge(from_node, to_node)

    return L


def _lg_undirected(G, selfloops=False, create_using=None):
    """Returns the line graph L of the (multi)graph G.

    Edges in G appear as nodes in L, represented as sorted tuples of the form
    (u,v), or (u,v,key) if G is a multigraph. A node in L corresponding to
    the edge {u,v} is connected to every node corresponding to an edge that
    involves u or v.

    Parameters
    ----------
    G : graph
        An undirected graph or multigraph.
    selfloops : bool
        If `True`, then self-loops are included in the line graph. If `False`,
        they are excluded.
    create_using : NetworkX graph constructor, optional (default=nx.Graph)
       Graph type to create. If graph instance, then cleared before populated.

    Notes
    -----
    The standard algorithm for line graphs of undirected graphs does not
    produce self-loops.

    """
    L = nx.empty_graph(0, create_using, default=G.__class__)

    # Graph specific functions for edges.
    get_edges = partial(G.edges, keys=True) if G.is_multigraph() else G.edges

    # Determine if we include self-loops or not.
    shift = 0 if selfloops else 1

    # Introduce numbering of nodes
    node_index = {n: i for i, n in enumerate(G)}

    # Lift canonical representation of nodes to edges in line graph
    edge_key_function = lambda edge: (node_index[edge[0]], node_index[edge[1]])

    edges = set()
    for u in G:
        # Label nodes as a sorted tuple of nodes in original graph.
        # Decide on representation of {u, v} as (u, v) or (v, u) depending on node_index.
        # -> This ensures a canonical representation and avoids comparing values of different types.
        nodes = [tuple(sorted(x[:2], key=node_index.get)) + x[2:] for x in get_edges(u)]

        if len(nodes) == 1:
            # Then the edge will be an isolated node in L.
            L.add_node(nodes[0])

        # Add a clique of `nodes` to graph. To prevent double adding edges,
        # especially important for multigraphs, we store the edges in
        # canonical form in a set.
        for i, a in enumerate(nodes):
            edges.update(
                [
                    tuple(sorted((a, b), key=edge_key_function))
                    for b in nodes[i + shift :]
                ]
            )

    L.add_edges_from(edges)
    return L


@not_implemented_for("directed")
@not_implemented_for("multigraph")
@nx._dispatchable(returns_graph=True)
def inverse_line_graph(G):
    """Returns the inverse line graph of graph G.

    If H is a graph, and G is the line graph of H, such that G = L(H).
    Then H is the inverse line graph of G.

    Not all graphs are line graphs and these do not have an inverse line graph.
    In these cases this function raises a NetworkXError.

    Parameters
    ----------
    G : graph
        A NetworkX Graph

    Returns
    -------
    H : graph
        The inverse line graph of G.

    Raises
    ------
    NetworkXNotImplemented
        If G is directed or a multigraph

    NetworkXError
        If G is not a line graph

    Notes
    -----
    This is an implementation of the Roussopoulos algorithm[1]_.

    If G consists of multiple components, then the algorithm doesn't work.
    You should invert every component separately:

    >>> K5 = nx.complete_graph(5)
    >>> P4 = nx.Graph([("a", "b"), ("b", "c"), ("c", "d")])
    >>> G = nx.union(K5, P4)
    >>> root_graphs = []
    >>> for comp in nx.connected_components(G):
    ...     root_graphs.append(nx.inverse_line_graph(G.subgraph(comp)))
    >>> len(root_graphs)
    2

    References
    ----------
    .. [1] Roussopoulos, N.D. , "A max {m, n} algorithm for determining the graph H from
       its line graph G", Information Processing Letters 2, (1973), 108--112, ISSN 0020-0190,
       `DOI link <https://doi.org/10.1016/0020-0190(73)90029-X>`_

    """
    if G.number_of_nodes() == 0:
        return nx.empty_graph(1)
    elif G.number_of_nodes() == 1:
        v = arbitrary_element(G)
        a = (v, 0)
        b = (v, 1)
        H = nx.Graph([(a, b)])
        return H
    elif G.number_of_nodes() > 1 and G.number_of_edges() == 0:
        msg = (
            "inverse_line_graph() doesn't work on an edgeless graph. "
            "Please use this function on each component separately."
        )
        raise nx.NetworkXError(msg)

    if nx.number_of_selfloops(G) != 0:
        msg = (
            "A line graph as generated by NetworkX has no selfloops, so G has no "
            "inverse line graph. Please remove the selfloops from G and try again."
        )
        raise nx.NetworkXError(msg)

    starting_cell = _select_starting_cell(G)
    P = _find_partition(G, starting_cell)
    # count how many times each vertex appears in the partition set
    P_count = {u: 0 for u in G.nodes}
    for p in P:
        for u in p:
            P_count[u] += 1

    if max(P_count.values()) > 2:
        msg = "G is not a line graph (vertex found in more than two partition cells)"
        raise nx.NetworkXError(msg)
    W = tuple((u,) for u in P_count if P_count[u] == 1)
    H = nx.Graph()
    H.add_nodes_from(P)
    H.add_nodes_from(W)
    for a, b in combinations(H.nodes, 2):
        if any(a_bit in b for a_bit in a):
            H.add_edge(a, b)
    return H


def _triangles(G, e):
    """Return list of all triangles containing edge e"""
    u, v = e
    if u not in G:
        raise nx.NetworkXError(f"Vertex {u} not in graph")
    if v not in G[u]:
        raise nx.NetworkXError(f"Edge ({u}, {v}) not in graph")
    triangle_list = []
    for x in G[u]:
        if x in G[v]:
            triangle_list.append((u, v, x))
    return triangle_list


def _odd_triangle(G, T):
    """Test whether T is an odd triangle in G

    Parameters
    ----------
    G : NetworkX Graph
    T : 3-tuple of vertices forming triangle in G

    Returns
    -------
    True is T is an odd triangle
    False otherwise

    Raises
    ------
    NetworkXError
        T is not a triangle in G

    Notes
    -----
    An odd triangle is one in which there exists another vertex in G which is
    adjacent to either exactly one or exactly all three of the vertices in the
    triangle.

    """
    for u in T:
        if u not in G.nodes():
            raise nx.NetworkXError(f"Vertex {u} not in graph")
    for e in list(combinations(T, 2)):
        if e[0] not in G[e[1]]:
            raise nx.NetworkXError(f"Edge ({e[0]}, {e[1]}) not in graph")

    T_nbrs = defaultdict(int)
    for t in T:
        for v in G[t]:
            if v not in T:
                T_nbrs[v] += 1
    return any(T_nbrs[v] in [1, 3] for v in T_nbrs)


def _find_partition(G, starting_cell):
    """Find a partition of the vertices of G into cells of complete graphs

    Parameters
    ----------
    G : NetworkX Graph
    starting_cell : tuple of vertices in G which form a cell

    Returns
    -------
    List of tuples of vertices of G

    Raises
    ------
    NetworkXError
        If a cell is not a complete subgraph then G is not a line graph
    """
    G_partition = G.copy()
    P = [starting_cell]  # partition set
    G_partition.remove_edges_from(list(combinations(starting_cell, 2)))
    # keep list of partitioned nodes which might have an edge in G_partition
    partitioned_vertices = list(starting_cell)
    while G_partition.number_of_edges() > 0:
        # there are still edges left and so more cells to be made
        u = partitioned_vertices.pop()
        deg_u = len(G_partition[u])
        if deg_u != 0:
            # if u still has edges then we need to find its other cell
            # this other cell must be a complete subgraph or else G is
            # not a line graph
            new_cell = [u] + list(G_partition[u])
            for u in new_cell:
                for v in new_cell:
                    if (u != v) and (v not in G_partition[u]):
                        msg = (
                            "G is not a line graph "
                            "(partition cell not a complete subgraph)"
                        )
                        raise nx.NetworkXError(msg)
            P.append(tuple(new_cell))
            G_partition.remove_edges_from(list(combinations(new_cell, 2)))
            partitioned_vertices += new_cell
    return P


def _select_starting_cell(G, starting_edge=None):
    """Select a cell to initiate _find_partition

    Parameters
    ----------
    G : NetworkX Graph
    starting_edge: an edge to build the starting cell from

    Returns
    -------
    Tuple of vertices in G

    Raises
    ------
    NetworkXError
        If it is determined that G is not a line graph

    Notes
    -----
    If starting edge not specified then pick an arbitrary edge - doesn't
    matter which. However, this function may call itself requiring a
    specific starting edge. Note that the r, s notation for counting
    triangles is the same as in the Roussopoulos paper cited above.
    """
    if starting_edge is None:
        e = arbitrary_element(G.edges())
    else:
        e = starting_edge
        if e[0] not in G.nodes():
            raise nx.NetworkXError(f"Vertex {e[0]} not in graph")
        if e[1] not in G[e[0]]:
            msg = f"starting_edge ({e[0]}, {e[1]}) is not in the Graph"
            raise nx.NetworkXError(msg)
    e_triangles = _triangles(G, e)
    r = len(e_triangles)
    if r == 0:
        # there are no triangles containing e, so the starting cell is just e
        starting_cell = e
    elif r == 1:
        # there is exactly one triangle, T, containing e. If other 2 edges
        # of T belong only to this triangle then T is starting cell
        T = e_triangles[0]
        a, b, c = T
        # ab was original edge so check the other 2 edges
        ac_edges = len(_triangles(G, (a, c)))
        bc_edges = len(_triangles(G, (b, c)))
        if ac_edges == 1:
            if bc_edges == 1:
                starting_cell = T
            else:
                return _select_starting_cell(G, starting_edge=(b, c))
        else:
            return _select_starting_cell(G, starting_edge=(a, c))
    else:
        # r >= 2 so we need to count the number of odd triangles, s
        s = 0
        odd_triangles = []
        for T in e_triangles:
            if _odd_triangle(G, T):
                s += 1
                odd_triangles.append(T)
        if r == 2 and s == 0:
            # in this case either triangle works, so just use T
            starting_cell = T
        elif r - 1 <= s <= r:
            # check if odd triangles containing e form complete subgraph
            triangle_nodes = set()
            for T in odd_triangles:
                for x in T:
                    triangle_nodes.add(x)

            for u in triangle_nodes:
                for v in triangle_nodes:
                    if u != v and (v not in G[u]):
                        msg = (
                            "G is not a line graph (odd triangles "
                            "do not form complete subgraph)"
                        )
                        raise nx.NetworkXError(msg)
            # otherwise then we can use this as the starting cell
            starting_cell = tuple(triangle_nodes)

        else:
            msg = (
                "G is not a line graph (incorrect number of "
                "odd triangles around starting edge)"
            )
            raise nx.NetworkXError(msg)
    return starting_cell