Python Graph.line_graph方法代码示例

# 需要导入模块: from sage.graphs.graph import Graph [as 别名]
# 或者: from sage.graphs.graph.Graph import line_graph [as 别名]
def is_line_graph(g, certificate = False):
    r"""
    Tests wether the graph is a line graph.

    INPUT:

    - ``certificate`` (boolean) -- whether to return a certificate along with
      the boolean result. Here is what happens when ``certificate = True``:

      - If the graph is not a line graph, the method returns a pair ``(b,
        subgraph)`` where ``b`` is ``False`` and ``subgraph`` is a subgraph
        isomorphic to one of the 9 forbidden induced subgraphs of a line graph.

      - If the graph is a line graph, the method returns a triple ``(b,R,isom)``
        where ``b`` is ``True``, ``R`` is a graph whose line graph is the graph
        given as input, and ``isom`` is a map associating an edge of ``R`` to
        each vertex of the graph.

    .. TODO::

        This method sequentially tests each of the forbidden subgraphs in order
        to know whether the graph is a line graph, which is a very slow
        method. It could eventually be replaced by
        :func:`~sage.graphs.line_graph.root_graph` when this method will not
        require an exponential time to run on general graphs anymore (see its
        documentation for more information on this problem)...  and if it can be
        improved to return negative certificates !

    .. NOTE::

        This method wastes a bit of time when the input graph is not
        connected. If you have performance in mind, it is probably better to
        only feed it with connected graphs only.

    .. SEEALSO::

        - The :mod:`line_graph <sage.graphs.line_graph>` module.

        - :meth:`~sage.graphs.graph_generators.GraphGenerators.line_graph_forbidden_subgraphs`
          -- the forbidden subgraphs of a line graph.

        - :meth:`~sage.graphs.generic_graph.GenericGraph.line_graph`

    EXAMPLES:

    A complete graph is always the line graph of a star::

        sage: graphs.CompleteGraph(5).is_line_graph()
        True

    The Petersen Graph not being claw-free, it is not a line
    graph::

        sage: graphs.PetersenGraph().is_line_graph()
        False

    This is indeed the subgraph returned::

        sage: C = graphs.PetersenGraph().is_line_graph(certificate = True)[1]
        sage: C.is_isomorphic(graphs.ClawGraph())
        True

    The house graph is a line graph::

        sage: g = graphs.HouseGraph()
        sage: g.is_line_graph()
        True

    But what is the graph whose line graph is the house ?::

        sage: is_line, R, isom = g.is_line_graph(certificate = True)
        sage: R.sparse6_string()
        ':DaHI~'
        sage: R.show()
        sage: isom
        {0: (0, 1), 1: (0, 2), 2: (1, 3), 3: (2, 3), 4: (3, 4)}

    TESTS:

    Disconnected graphs::

        sage: g = 2*graphs.CycleGraph(3)
        sage: gl = g.line_graph().relabel(inplace = False)
        sage: new_g = gl.is_line_graph(certificate = True)[1]
        sage: g.line_graph().is_isomorphic(gl)
        True
    """
    g._scream_if_not_simple()
    from sage.graphs.graph_generators import graphs

    for fg in graphs.line_graph_forbidden_subgraphs():
        h = g.subgraph_search(fg, induced = True)
        if h is not None:
            if certificate:
                return (False,h)
            else:
                return False

    if not certificate:
        return True
#.........这里部分代码省略.........

# 需要导入模块: from sage.graphs.graph import Graph [as 别名]
# 或者: from sage.graphs.graph.Graph import line_graph [as 别名]
def root_graph(g, verbose = False):
    r"""
    Computes the root graph corresponding to the given graph

    See the documentation of :mod:`sage.graphs.line_graph` to know how it works.

    INPUT:

    - ``g`` -- a graph

    - ``verbose`` (boolean) -- display some information about what is happening
      inside of the algorithm.

    .. NOTE::

        It is best to use this code through
        :meth:`~sage.graphs.graph.Graph.is_line_graph`, which first checks that
        the graph is indeed a line graph, and deals with the disconnected
        case. But if you are sure of yourself, dig in !

    .. WARNING::

        * This code assumes that the graph is connected.

        * If the graph is *not* a line graph, this implementation will take a
          loooooong time to run. Its first step is to enumerate all maximal
          cliques, and that can take a while for general graphs. As soon as
          there is a way to iterate over maximal cliques without first building
          the (long) list of them this implementation can be updated, and will
          deal reasonably with non-line graphs too !

    TESTS:

    All connected graphs on 6 vertices::

        sage: from sage.graphs.line_graph import root_graph
        sage: def test(g):
        ....:    gl = g.line_graph(labels = False)
        ....:    d=root_graph(gl)
        sage: for i,g in enumerate(graphs(6)): # long time
        ....:   if not g.is_connected():       # long time
        ....:     continue                     # long time
        ....:   test(g)                        # long time

    Non line-graphs::

        sage: root_graph(graphs.PetersenGraph())
        Traceback (most recent call last):
        ...
        ValueError: This graph is not a line graph !

    Small corner-cases::

        sage: from sage.graphs.line_graph import root_graph
        sage: root_graph(graphs.CompleteGraph(3))
        (Complete bipartite graph: Graph on 4 vertices, {0: (0, 1), 1: (0, 2), 2: (0, 3)})
        sage: root_graph(graphs.OctahedralGraph())
        (Complete graph: Graph on 4 vertices, {0: (0, 1), 1: (0, 2), 2: (0, 3), 3: (1, 2), 4: (1, 3), 5: (2, 3)})
        sage: root_graph(graphs.DiamondGraph())
        (Graph on 4 vertices, {0: (0, 3), 1: (0, 1), 2: (0, 2), 3: (1, 2)})
        sage: root_graph(graphs.WheelGraph(5))
        (Diamond Graph: Graph on 4 vertices, {0: (1, 2), 1: (0, 1), 2: (0, 2), 3: (2, 3), 4: (1, 3)})
    """
    from sage.graphs.digraph import DiGraph

    if isinstance(g, DiGraph):
        raise ValueError("g cannot be a DiGraph !")
    if g.has_multiple_edges():
        raise ValueError("g cannot have multiple edges !")
    if not g.is_connected():
        raise ValueError("g is not connected !")

    # Complete Graph ?
    if g.is_clique():
        from sage.graphs.generators.basic import CompleteBipartiteGraph
        return (CompleteBipartiteGraph(1,g.order()),
                {v : (0,1+i) for i,v in enumerate(g)})

    # Diamond Graph ?
    elif g.order() == 4 and g.size() == 5:
        from sage.graphs.graph import Graph
        root = Graph([(0,1),(1,2),(2,0),(0,3)])
        return (root,
                g.is_isomorphic(root.line_graph(labels = False), certificate = True)[1])

    # Wheel on 5 vertices ?
    elif g.order() == 5 and g.size() == 8 and min(g.degree()) == 3:
        from sage.graphs.generators.basic import DiamondGraph
        root = DiamondGraph()
        return (root,
                g.is_isomorphic(root.line_graph(labels = False), certificate = True)[1])

    # Octahedron ?
    elif g.order() == 6 and g.size() == 12 and g.is_regular(k=4):
        from sage.graphs.generators.platonic_solids import OctahedralGraph
        if g.is_isomorphic(OctahedralGraph()):
            from sage.graphs.generators.basic import CompleteGraph
            root = CompleteGraph(4)
            return (root,
                    g.is_isomorphic(root.line_graph(labels = False), certificate = True)[1])
#.........这里部分代码省略.........

# 需要导入模块: from sage.graphs.graph import Graph [as 别名]
# 或者: from sage.graphs.graph.Graph import line_graph [as 别名]
def is_line_graph(g, certificate=False):
    r"""
    Tests wether the graph is a line graph.

    INPUT:

    - ``certificate`` (boolean) -- whether to return a certificate along with
      the boolean result. Here is what happens when ``certificate = True``:

      - If the graph is not a line graph, the method returns a pair ``(b,
        subgraph)`` where ``b`` is ``False`` and ``subgraph`` is a subgraph
        isomorphic to one of the 9 forbidden induced subgraphs of a line graph.

      - If the graph is a line graph, the method returns a triple ``(b,R,isom)``
        where ``b`` is ``True``, ``R`` is a graph whose line graph is the graph
        given as input, and ``isom`` is a map associating an edge of ``R`` to
        each vertex of the graph.

    .. NOTE::

        This method wastes a bit of time when the input graph is not connected.
        If you have performance in mind, it is probably better to only feed it
        with connected graphs only.

    .. SEEALSO::

        - The :mod:`line_graph <sage.graphs.line_graph>` module.

        - :meth:`~sage.graphs.graph_generators.GraphGenerators.line_graph_forbidden_subgraphs`
          -- the forbidden subgraphs of a line graph.

        - :meth:`~sage.graphs.generic_graph.GenericGraph.line_graph`

    EXAMPLES:

    A complete graph is always the line graph of a star::

        sage: graphs.CompleteGraph(5).is_line_graph()
        True

    The Petersen Graph not being claw-free, it is not a line
    graph::

        sage: graphs.PetersenGraph().is_line_graph()
        False

    This is indeed the subgraph returned::

        sage: C = graphs.PetersenGraph().is_line_graph(certificate=True)[1]
        sage: C.is_isomorphic(graphs.ClawGraph())
        True

    The house graph is a line graph::

        sage: g = graphs.HouseGraph()
        sage: g.is_line_graph()
        True

    But what is the graph whose line graph is the house ?::

        sage: is_line, R, isom = g.is_line_graph(certificate=True)
        sage: R.sparse6_string()
        ':DaHI~'
        sage: R.show()
        sage: isom
        {0: (0, 1), 1: (0, 2), 2: (1, 3), 3: (2, 3), 4: (3, 4)}

    TESTS:

    Disconnected graphs::

        sage: g = 2 * graphs.CycleGraph(3)
        sage: gl = g.line_graph().relabel(inplace=False)
        sage: new_g = gl.is_line_graph(certificate=True)[1]
        sage: g.line_graph().is_isomorphic(gl)
        True
    """
    g._scream_if_not_simple()

    def get_certificate(gg):
        """
        Assuming that gg is not a line graph, return a forbidden induced
        subgraph.
        """
        from sage.graphs.graph_generators import graphs
        for fg in graphs.line_graph_forbidden_subgraphs():
            h = gg.subgraph_search(fg, induced=True)
            if h is not None:
                return h

    if g.is_connected():
        try:
            R, isom = root_graph(g)
            if certificate:
                return True, R, isom
            else:
                return True
        except ValueError:
            # g is not a line graph
            if certificate:
#.........这里部分代码省略.........