Python networkx.is_chordal() Examples

def get_clique_tree(nodes, edges):
    """Given a set of int nodes i and edges (i,j), returns an nx.Graph object G
    which is a clique tree, where:
        - G.node[i]['members'] contains the set of original nodes in the ith
            maximal clique
        - G[i][j]['members'] contains the set of original nodes in the seperator
            set between maximal cliques i and j

    Note: This method is currently only implemented for chordal graphs; TODO:
    add a step to triangulate non-chordal graphs.
    """
    # Form the original graph G1
    G1 = nx.Graph()
    G1.add_nodes_from(nodes)
    G1.add_edges_from(edges)

    # Check if graph is chordal
    # TODO: Add step to triangulate graph if not
    if not nx.is_chordal(G1):
        raise NotImplementedError("Graph triangulation not implemented.")

    # Create maximal clique graph G2
    # Each node is a maximal clique C_i
    # Let w = |C_i \cap C_j|; C_i, C_j have an edge with weight w if w > 0
    G2 = nx.Graph()
    for i, c in enumerate(nx.chordal_graph_cliques(G1)):
        G2.add_node(i, members=c)
    for i in G2.nodes:
        for j in G2.nodes:
            S = G2.node[i]["members"].intersection(G2.node[j]["members"])
            w = len(S)
            if w > 0:
                G2.add_edge(i, j, weight=w, members=S)

    # Return a minimum spanning tree of G2
    return nx.minimum_spanning_tree(G2) 

def chordal_graph_cliques(G):
    """Returns the set of maximal cliques of a chordal graph.
    
    The algorithm breaks the graph in connected components and performs a 
    maximum cardinality search in each component to get the cliques.
    
    Parameters
    ----------
    G : graph
      A NetworkX graph 
    
    Returns
    -------
    cliques : A set containing the maximal cliques in G.
    
    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. 
        If the input graph is an instance of one of these classes, a
        NetworkXError is raised.
        The algorithm can only be applied to chordal graphs. If the
        input graph is found to be non-chordal, a NetworkXError is raised.
        
    Examples
    --------
    >>> import networkx as nx
    >>> e= [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
    >>> G = nx.Graph(e)
    >>> G.add_node(9)    
    >>> setlist = nx.chordal_graph_cliques(G)
    """
    if not is_chordal(G):
        raise nx.NetworkXError("Input graph is not chordal.")
    
    cliques = set()
    for C in nx.connected.connected_component_subgraphs(G):
        cliques |= _connected_chordal_graph_cliques(C)
        
    return cliques 

def test_is_chordal(self):
        assert_false(nx.is_chordal(self.non_chordal_G))
        assert_true(nx.is_chordal(self.chordal_G))
        assert_true(nx.is_chordal(self.connected_chordal_G))
        assert_true(nx.is_chordal(nx.complete_graph(3)))
        assert_true(nx.is_chordal(nx.cycle_graph(3)))
        assert_false(nx.is_chordal(nx.cycle_graph(5))) 

def is_chordal(edge_list):
	"""
	Check if the graph is chordal/triangulated.

	Parameters
	----------
	*edge_list* : a list of lists (optional)
	The edges to check (if not self.E)

	Returns
	-------
	*nx.is_chordal(G)* : a boolean
	Whether the graph is chordal or not

	Effects
	-------
	None

	Notes
	-----
	Again, do we need networkx for this? Eventually should
	write this check on our own.

	"""
	G = nx.Graph(list(edge_list))
	return nx.is_chordal(G) 

def test_is_chordal(self):
        assert_false(nx.is_chordal(self.non_chordal_G))
        assert_true(nx.is_chordal(self.chordal_G))
        assert_true(nx.is_chordal(self.connected_chordal_G))
        assert_true(nx.is_chordal(nx.complete_graph(3)))
        assert_true(nx.is_chordal(nx.cycle_graph(3)))
        assert_false(nx.is_chordal(nx.cycle_graph(5))) 

def chordal_graph_cliques(G):
    """Returns the set of maximal cliques of a chordal graph.

    The algorithm breaks the graph in connected components and performs a
    maximum cardinality search in each component to get the cliques.

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

    Returns
    -------
    cliques : A set containing the maximal cliques in G.

    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
        If the input graph is an instance of one of these classes, a
        :exc:`NetworkXError` is raised.
        The algorithm can only be applied to chordal graphs. If the
        input graph is found to be non-chordal, a :exc:`NetworkXError` is raised.

    Examples
    --------
    >>> import networkx as nx
    >>> e= [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
    >>> G = nx.Graph(e)
    >>> G.add_node(9)
    >>> setlist = nx.chordal_graph_cliques(G)
    """
    if not is_chordal(G):
        raise nx.NetworkXError("Input graph is not chordal.")

    cliques = set()
    for C in nx.connected.connected_component_subgraphs(G):
        cliques |= _connected_chordal_graph_cliques(C)

    return cliques 

def test_is_chordal(self):
        assert_false(nx.is_chordal(self.non_chordal_G))
        assert_true(nx.is_chordal(self.chordal_G))
        assert_true(nx.is_chordal(self.connected_chordal_G))
        assert_true(nx.is_chordal(nx.complete_graph(3)))
        assert_true(nx.is_chordal(nx.cycle_graph(3)))
        assert_false(nx.is_chordal(nx.cycle_graph(5))) 

def chordal_graph_cliques(G):
    """Returns the set of maximal cliques of a chordal graph.

    The algorithm breaks the graph in connected components and performs a
    maximum cardinality search in each component to get the cliques.

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

    Returns
    -------
    cliques : A set containing the maximal cliques in G.

    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
        If the input graph is an instance of one of these classes, a
        :exc:`NetworkXError` is raised.
        The algorithm can only be applied to chordal graphs. If the
        input graph is found to be non-chordal, a :exc:`NetworkXError` is raised.

    Examples
    --------
    >>> import networkx as nx
    >>> e= [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
    >>> G = nx.Graph(e)
    >>> G.add_node(9)
    >>> setlist = nx.chordal_graph_cliques(G)
    """
    if not is_chordal(G):
        raise nx.NetworkXError("Input graph is not chordal.")

    cliques = set()
    for C in nx.connected.connected_component_subgraphs(G):
        cliques |= _connected_chordal_graph_cliques(C)

    return cliques 

def get_clique_tree(nodes: Iterable[int], edges: List[Tuple[int, int]]) -> nx.Graph:
    """
    Given a set of int nodes i and edges (i,j), returns a clique tree.

    Clique tree is an object G for which:
    - G.node[i]['members'] contains the set of original nodes in the ith
        maximal clique
    - G[i][j]['members'] contains the set of original nodes in the seperator
        set between maximal cliques i and j

    Note: This method is currently only implemented for chordal graphs; TODO:
    add a step to triangulate non-chordal graphs.

    Parameters
    ----------
    nodes
        A list of nodes indices
    edges
        A list of tuples, where each tuple has indices for connected nodes

    Returns
    -------
    networkx.Graph
        An object G representing clique tree
    """
    # Form the original graph G1
    G1 = nx.Graph()
    G1.add_nodes_from(nodes)
    G1.add_edges_from(edges)

    # Check if graph is chordal
    # TODO: Add step to triangulate graph if not
    if not nx.is_chordal(G1):
        raise NotImplementedError("Graph triangulation not implemented.")

    # Create maximal clique graph G2
    # Each node is a maximal clique C_i
    # Let w = |C_i \cap C_j|; C_i, C_j have an edge with weight w if w > 0
    G2 = nx.Graph()
    for i, c in enumerate(nx.chordal_graph_cliques(G1)):
        G2.add_node(i, members=c)
    for i in G2.nodes:
        for j in G2.nodes:
            S = G2.node[i]["members"].intersection(G2.node[j]["members"])
            w = len(S)
            if w > 0:
                G2.add_edge(i, j, weight=w, members=S)

    # Return a minimum spanning tree of G2
    return nx.minimum_spanning_tree(G2) 

def is_chordal(G):
    """Checks whether G is a chordal graph.

    A graph is chordal if every cycle of length at least 4 has a chord
    (an edge joining two nodes not adjacent in the cycle).

    Parameters
    ----------
    G : graph  
      A NetworkX graph.  

    Returns
    -------
    chordal : bool
      True if G is a chordal graph and False otherwise.
    
    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. 
        If the input graph is an instance of one of these classes, a
        NetworkXError is raised.
        
    Examples
    --------
    >>> import networkx as nx
    >>> e=[(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)]
    >>> G=nx.Graph(e)
    >>> nx.is_chordal(G)
    True
   
    Notes
    -----
    The routine tries to go through every node following maximum cardinality 
    search. It returns False when it finds that the separator for any node 
    is not a clique.  Based on the algorithms in [1]_.

    References
    ----------
    .. [1] R. E. Tarjan and M. Yannakakis, Simple linear-time algorithms 
       to test chordality of graphs, test acyclicity of hypergraphs, and 
       selectively reduce acyclic hypergraphs, SIAM J. Comput., 13 (1984), 
       pp. 566–579.
    """
    if G.is_directed():
        raise nx.NetworkXError('Directed graphs not supported')
    if G.is_multigraph():
        raise nx.NetworkXError('Multiply connected graphs not supported.')
    if len(_find_chordality_breaker(G))==0:
        return True
    else:
        return False 

def chordal_graph_treewidth(G):
    """Returns the treewidth of the chordal graph G.
 
    Parameters
    ----------
    G : graph
      A NetworkX graph 
    
    Returns
    -------
    treewidth : int
	The size of the largest clique in the graph minus one.
    
    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph. 
        If the input graph is an instance of one of these classes, a
        NetworkXError is raised.
        The algorithm can only be applied to chordal graphs. If
        the input graph is found to be non-chordal, a NetworkXError is raised.
        
    Examples
    --------
    >>> import networkx as nx
    >>> e = [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
    >>> G = nx.Graph(e)
    >>> G.add_node(9)    
    >>> nx.chordal_graph_treewidth(G)
    3

    References
    ----------
    .. [1] http://en.wikipedia.org/wiki/Tree_decomposition#Treewidth
    """
    if not is_chordal(G):
        raise nx.NetworkXError("Input graph is not chordal.")
    
    max_clique = -1
    for clique in nx.chordal_graph_cliques(G):
        max_clique = max(max_clique,len(clique))
    return max_clique - 1 

def is_chordal(G):
    """Checks whether G is a chordal graph.

    A graph is chordal if every cycle of length at least 4 has a chord
    (an edge joining two nodes not adjacent in the cycle).

    Parameters
    ----------
    G : graph
      A NetworkX graph.

    Returns
    -------
    chordal : bool
      True if G is a chordal graph and False otherwise.

    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
        If the input graph is an instance of one of these classes, a
        :exc:`NetworkXError` is raised.

    Examples
    --------
    >>> import networkx as nx
    >>> e=[(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)]
    >>> G=nx.Graph(e)
    >>> nx.is_chordal(G)
    True

    Notes
    -----
    The routine tries to go through every node following maximum cardinality
    search. It returns False when it finds that the separator for any node
    is not a clique.  Based on the algorithms in [1]_.

    References
    ----------
    .. [1] R. E. Tarjan and M. Yannakakis, Simple linear-time algorithms
       to test chordality of graphs, test acyclicity of hypergraphs, and
       selectively reduce acyclic hypergraphs, SIAM J. Comput., 13 (1984),
       pp. 566–579.
    """
    if G.is_directed():
        raise nx.NetworkXError('Directed graphs not supported')
    if G.is_multigraph():
        raise nx.NetworkXError('Multiply connected graphs not supported.')
    if len(_find_chordality_breaker(G)) == 0:
        return True
    else:
        return False 

def chordal_graph_treewidth(G):
    """Returns the treewidth of the chordal graph G.

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

    Returns
    -------
    treewidth : int
        The size of the largest clique in the graph minus one.

    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
        If the input graph is an instance of one of these classes, a
        :exc:`NetworkXError` is raised.
        The algorithm can only be applied to chordal graphs. If
        the input graph is found to be non-chordal, a :exc:`NetworkXError` is raised.

    Examples
    --------
    >>> import networkx as nx
    >>> e = [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
    >>> G = nx.Graph(e)
    >>> G.add_node(9)
    >>> nx.chordal_graph_treewidth(G)
    3

    References
    ----------
    .. [1] https://en.wikipedia.org/wiki/Tree_decomposition#Treewidth
    """
    if not is_chordal(G):
        raise nx.NetworkXError("Input graph is not chordal.")

    max_clique = -1
    for clique in nx.chordal_graph_cliques(G):
        max_clique = max(max_clique, len(clique))
    return max_clique - 1 

def is_chordal(G):
    """Checks whether G is a chordal graph.

    A graph is chordal if every cycle of length at least 4 has a chord
    (an edge joining two nodes not adjacent in the cycle).

    Parameters
    ----------
    G : graph
      A NetworkX graph.

    Returns
    -------
    chordal : bool
      True if G is a chordal graph and False otherwise.

    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
        If the input graph is an instance of one of these classes, a
        :exc:`NetworkXError` is raised.

    Examples
    --------
    >>> import networkx as nx
    >>> e=[(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)]
    >>> G=nx.Graph(e)
    >>> nx.is_chordal(G)
    True

    Notes
    -----
    The routine tries to go through every node following maximum cardinality
    search. It returns False when it finds that the separator for any node
    is not a clique.  Based on the algorithms in [1]_.

    References
    ----------
    .. [1] R. E. Tarjan and M. Yannakakis, Simple linear-time algorithms
       to test chordality of graphs, test acyclicity of hypergraphs, and
       selectively reduce acyclic hypergraphs, SIAM J. Comput., 13 (1984),
       pp. 566–579.
    """
    if G.is_directed():
        raise nx.NetworkXError('Directed graphs not supported')
    if G.is_multigraph():
        raise nx.NetworkXError('Multiply connected graphs not supported.')
    if len(_find_chordality_breaker(G)) == 0:
        return True
    else:
        return False 

def chordal_graph_treewidth(G):
    """Returns the treewidth of the chordal graph G.

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

    Returns
    -------
    treewidth : int
        The size of the largest clique in the graph minus one.

    Raises
    ------
    NetworkXError
        The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
        If the input graph is an instance of one of these classes, a
        :exc:`NetworkXError` is raised.
        The algorithm can only be applied to chordal graphs. If
        the input graph is found to be non-chordal, a :exc:`NetworkXError` is raised.

    Examples
    --------
    >>> import networkx as nx
    >>> e = [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
    >>> G = nx.Graph(e)
    >>> G.add_node(9)
    >>> nx.chordal_graph_treewidth(G)
    3

    References
    ----------
    .. [1] https://en.wikipedia.org/wiki/Tree_decomposition#Treewidth
    """
    if not is_chordal(G):
        raise nx.NetworkXError("Input graph is not chordal.")

    max_clique = -1
    for clique in nx.chordal_graph_cliques(G):
        max_clique = max(max_clique, len(clique))
    return max_clique - 1