text: """Generators for some classic graphs.
  
  The typical graph builder function is called as follows:
  
  >>> G = nx.complete_graph(100)
  
  returning the complete graph on n nodes labeled 0, .., 99
  as a simple graph. Except for `empty_graph`, all the functions
  in this module return a Graph class (i.e. a simple, undirected graph).
  
  """
  
  import itertools
  
  import networkx as nx
  from networkx.classes import Graph
  from networkx.exception import NetworkXError
  from networkx.utils import nodes_or_number
  from networkx.utils import pairwise
  
  __all__ = [
      "balanced_tree",
      "barbell_graph",
      "binomial_tree",
      "complete_graph",
      "complete_multipartite_graph",
      "circular_ladder_graph",
      "circulant_graph",
      "cycle_graph",
      "dorogovtsev_goltsev_mendes_graph",
      "empty_graph",
      "full_rary_tree",
      "ladder_graph",
      "lollipop_graph",
      "null_graph",
      "path_graph",
      "star_graph",
      "trivial_graph",
      "turan_graph",
      "wheel_graph",
  ]
  
  
  # -------------------------------------------------------------------
  #   Some Classic Graphs
  # -------------------------------------------------------------------
  
  
  def _tree_edges(n, r):
      if n == 0:
          return
      # helper function for trees
      # yields edges in rooted tree at 0 with n nodes and branching ratio r
      nodes = iter(range(n))
      parents = [next(nodes)]  # stack of max length r
      while parents:
          source = parents.pop(0)
          for i in range(r):
              try:
                  target = next(nodes)
                  parents.append(target)
                  yield source, target
              except StopIteration:
                  break
  
  
  def full_rary_tree(r, n, create_using=None):
      """Creates a full r-ary tree of `n` nodes.
  
      Sometimes called a k-ary, n-ary, or m-ary tree.
      "... all non-leaf nodes have exactly r children and all levels
      are full except for some rightmost position of the bottom level
      (if a leaf at the bottom level is missing, then so are all of the
      leaves to its right." [1]_
  
      Parameters
      ----------
      r : int
          branching factor of the tree
      n : int
          Number of nodes in the tree
      create_using : NetworkX graph constructor, optional (default=nx.Graph)
         Graph type to create. If graph instance, then cleared before populated.
  
      Returns
      -------
      G : networkx Graph
          An r-ary tree with n nodes
  
      References
      ----------
      .. [1] An introduction to data structures and algorithms,
             James Andrew Storer,  Birkhauser Boston 2001, (page 225).
      """
      G = empty_graph(n, create_using)
      G.add_edges_from(_tree_edges(n, r))
      return G
  
  
  def balanced_tree(r, h, create_using=None):
      """Returns the perfectly balanced `r`-ary tree of height `h`.
  
      Parameters
      ----------
      r : int
          Branching factor of the tree; each node will have `r`
          children.
  
      h : int
          Height of the tree.
  
      create_using : NetworkX graph constructor, optional (default=nx.Graph)
         Graph type to create. If graph instance, then cleared before populated.
  
      Returns
      -------
      G : NetworkX graph
          A balanced `r`-ary tree of height `h`.
  
      Notes
      -----
      This is the rooted tree where all leaves are at distance `h` from
      the root. The root has degree `r` and all other internal nodes
      have degree `r + 1`.
  
      Node labels are integers, starting from zero.
  
      A balanced tree is also known as a *complete r-ary tree*.
  
      """
      # The number of nodes in the balanced tree is `1 + r + ... + r^h`,
      # which is computed by using the closed-form formula for a geometric
      # sum with ratio `r`. In the special case that `r` is 1, the number
      # of nodes is simply `h + 1` (since the tree is actually a path
      # graph).
      if r == 1:
          n = h + 1
      else:
          # This must be an integer if both `r` and `h` are integers. If
          # they are not, we force integer division anyway.
          n = (1 - r ** (h + 1)) // (1 - r)
      return full_rary_tree(r, n, create_using=create_using)
  
  
  def barbell_graph(m1, m2, create_using=None):
      """Returns the Barbell Graph: two complete graphs connected by a path.
  
      For $m1 > 1$ and $m2 >= 0$.
  
      Two identical complete graphs $K_{m1}$ form the left and right bells,
      and are connected by a path $P_{m2}$.
  
      The `2*m1+m2`  nodes are numbered
          `0, ..., m1-1` for the left barbell,
          `m1, ..., m1+m2-1` for the path,
          and `m1+m2, ..., 2*m1+m2-1` for the right barbell.
  
      The 3 subgraphs are joined via the edges `(m1-1, m1)` and
      `(m1+m2-1, m1+m2)`. If `m2=0`, this is merely two complete
      graphs joined together.
  
      This graph is an extremal example in David Aldous
      and Jim Fill's e-text on Random Walks on Graphs.
  
      """
      if m1 < 2:
          raise NetworkXError("Invalid graph description, m1 should be >=2")
      if m2 < 0:
          raise NetworkXError("Invalid graph description, m2 should be >=0")
  
      # left barbell
      G = complete_graph(m1, create_using)
      if G.is_directed():
          raise NetworkXError("Directed Graph not supported")
  
      # connecting path
      G.add_nodes_from(range(m1, m1 + m2 - 1))
      if m2 > 1:
          G.add_edges_from(pairwise(range(m1, m1 + m2)))
      # right barbell
      G.add_edges_from(
          (u, v) for u in range(m1 + m2, 2 * m1 + m2) for v in range(u + 1, 2 * m1 + m2)
      )
      # connect it up
      G.add_edge(m1 - 1, m1)
      if m2 > 0:
          G.add_edge(m1 + m2 - 1, m1 + m2)
      return G
  
  
  def binomial_tree(n):
      """Returns the Binomial Tree of order n.
  
      The binomial tree of order 0 consists of a single node. A binomial tree of order k
      is defined recursively by linking two binomial trees of order k-1: the root of one is
      the leftmost child of the root of the other.
  
      Parameters
      ----------
      n : int
          Order of the binomial tree.
  
      Returns
      -------
      G : NetworkX graph
          A binomial tree of $2^n$ nodes and $2^n - 1$ edges.
  
      """
      G = nx.empty_graph(1)
      N = 1
      for i in range(n):
          edges = [(u + N, v + N) for (u, v) in G.edges]
          G.add_edges_from(edges)
          G.add_edge(0, N)
          N *= 2
      return G
  
  
  @nodes_or_number(0)
  def complete_graph(n, create_using=None):
      """Return the complete graph `K_n` with n nodes.
  
      A complete graph on `n` nodes means that all pairs
      of distinct nodes have an edge connecting them.
  
      Parameters
      ----------
      n : int or iterable container of nodes
          If n is an integer, nodes are from range(n).
          If n is a container of nodes, those nodes appear in the graph.
      create_using : NetworkX graph constructor, optional (default=nx.Graph)
         Graph type to create. If graph instance, then cleared before populated.
  
      Examples
      --------
      >>> G = nx.complete_graph(9)
      >>> len(G)
      9
      >>> G.size()
      36
      >>> G = nx.complete_graph(range(11, 14))
      >>> list(G.nodes())
      [11, 12, 13]
      >>> G = nx.complete_graph(4, nx.DiGraph())
      >>> G.is_directed()
      True
  
      """
      n_name, nodes = n
      G = empty_graph(n_name, create_using)
      if len(nodes) > 1:
          if G.is_directed():
              edges = itertools.permutations(nodes, 2)
          else:
              edges = itertools.combinations(nodes, 2)
          G.add_edges_from(edges)
      return G
  
  
  def circular_ladder_graph(n, create_using=None):
      """Returns the circular ladder graph $CL_n$ of length n.
  
      $CL_n$ consists of two concentric n-cycles in which
      each of the n pairs of concentric nodes are joined by an edge.
  
      Node labels are the integers 0 to n-1
  
      """
      G = ladder_graph(n, create_using)
      G.add_edge(0, n - 1)
      G.add_edge(n, 2 * n - 1)
      return G
  
  
  def circulant_graph(n, offsets, create_using=None):
      """Returns the circulant graph $Ci_n(x_1, x_2, ..., x_m)$ with $n$ nodes.
  
      The circulant graph $Ci_n(x_1, ..., x_m)$ consists of $n$ nodes $0, ..., n-1$
      such that node $i$ is connected to nodes $(i + x) \mod n$ and $(i - x) \mod n$
      for all $x$ in $x_1, ..., x_m$. Thus $Ci_n(1)$ is a cycle graph.
  
      Parameters
      ----------
      n : integer
          The number of nodes in the graph.
      offsets : list of integers
          A list of node offsets, $x_1$ up to $x_m$, as described above.
      create_using : NetworkX graph constructor, optional (default=nx.Graph)
         Graph type to create. If graph instance, then cleared before populated.
  
      Returns
      -------
      NetworkX Graph of type create_using
  
      Examples
      --------
      Many well-known graph families are subfamilies of the circulant graphs;
      for example, to create the cycle graph on n points, we connect every
      node to nodes on either side (with offset plus or minus one). For n = 10,
  
      >>> G = nx.circulant_graph(10, [1])
      >>> edges = [
      ...     (0, 9),
      ...     (0, 1),
      ...     (1, 2),
      ...     (2, 3),
      ...     (3, 4),
      ...     (4, 5),
      ...     (5, 6),
      ...     (6, 7),
      ...     (7, 8),
      ...     (8, 9),
      ... ]
      ...
      >>> sorted(edges) == sorted(G.edges())
      True
  
      Similarly, we can create the complete graph
      on 5 points with the set of offsets [1, 2]:
  
      >>> G = nx.circulant_graph(5, [1, 2])
      >>> edges = [
      ...     (0, 1),
      ...     (0, 2),
      ...     (0, 3),
      ...     (0, 4),
      ...     (1, 2),
      ...     (1, 3),
      ...     (1, 4),
      ...     (2, 3),
      ...     (2, 4),
      ...     (3, 4),
      ... ]
      ...
      >>> sorted(edges) == sorted(G.edges())
      True
  
      """
      G = empty_graph(n, create_using)
      for i in range(n):
          for j in offsets:
              G.add_edge(i, (i - j) % n)
              G.add_edge(i, (i + j) % n)
      return G
  
  
  @nodes_or_number(0)
  def cycle_graph(n, create_using=None):
      """Returns the cycle graph $C_n$ of cyclically connected nodes.
  
      $C_n$ is a path with its two end-nodes connected.
  
      Parameters
      ----------
      n : int or iterable container of nodes
          If n is an integer, nodes are from `range(n)`.
          If n is a container of nodes, those nodes appear in the graph.
      create_using : NetworkX graph constructor, optional (default=nx.Graph)
         Graph type to create. If graph instance, then cleared before populated.
  
      Notes
      -----
      If create_using is directed, the direction is in increasing order.
  
      """
      n_orig, nodes = n
      G = empty_graph(nodes, create_using)
      G.add_edges_from(pairwise(nodes))
      G.add_edge(nodes[-1], nodes[0])
      return G
  
  
  def dorogovtsev_goltsev_mendes_graph(n, create_using=None):
      """Returns the hierarchically constructed Dorogovtsev-Goltsev-Mendes graph.
  
      n is the generation.
      See: arXiv:/cond-mat/0112143 by Dorogovtsev, Goltsev and Mendes.
  
      """
      G = empty_graph(0, create_using)
      if G.is_directed():
          raise NetworkXError("Directed Graph not supported")
      if G.is_multigraph():
          raise NetworkXError("Multigraph not supported")
  
      G.add_edge(0, 1)
      if n == 0:
          return G
      new_node = 2  # next node to be added
      for i in range(1, n + 1):  # iterate over number of generations.
          last_generation_edges = list(G.edges())
          number_of_edges_in_last_generation = len(last_generation_edges)
          for j in range(0, number_of_edges_in_last_generation):
              G.add_edge(new_node, last_generation_edges[j][0])
              G.add_edge(new_node, last_generation_edges[j][1])
              new_node += 1
      return G
  
  
  @nodes_or_number(0)
  def empty_graph(n=0, create_using=None, default=Graph):
      """Returns the empty graph with n nodes and zero edges.
  
      Parameters
      ----------
      n : int or iterable container of nodes (default = 0)
          If n is an integer, nodes are from `range(n)`.
          If n is a container of nodes, those nodes appear in the graph.
      create_using : Graph Instance, Constructor or None
          Indicator of type of graph to return.
          If a Graph-type instance, then clear and use it.
          If None, use the `default` constructor.
          If a constructor, call it to create an empty graph.
      default : Graph constructor (optional, default = nx.Graph)
          The constructor to use if create_using is None.
          If None, then nx.Graph is used.
          This is used when passing an unknown `create_using` value
          through your home-grown function to `empty_graph` and
          you want a default constructor other than nx.Graph.
  
      Examples
      --------
      >>> G = nx.empty_graph(10)
      >>> G.number_of_nodes()
      10
      >>> G.number_of_edges()
      0
      >>> G = nx.empty_graph("ABC")
      >>> G.number_of_nodes()
      3
      >>> sorted(G)
      ['A', 'B', 'C']
  
      Notes
      -----
      The variable create_using should be a Graph Constructor or a
      "graph"-like object. Constructors, e.g. `nx.Graph` or `nx.MultiGraph`
      will be used to create the returned graph. "graph"-like objects
      will be cleared (nodes and edges will be removed) and refitted as
      an empty "graph" with nodes specified in n. This capability
      is useful for specifying the class-nature of the resulting empty
      "graph" (i.e. Graph, DiGraph, MyWeirdGraphClass, etc.).
  
      The variable create_using has three main uses:
      Firstly, the variable create_using can be used to create an
      empty digraph, multigraph, etc.  For example,
  
      >>> n = 10
      >>> G = nx.empty_graph(n, create_using=nx.DiGraph)
  
      will create an empty digraph on n nodes.
  
      Secondly, one can pass an existing graph (digraph, multigraph,
      etc.) via create_using. For example, if G is an existing graph
      (resp. digraph, multigraph, etc.), then empty_graph(n, create_using=G)
      will empty G (i.e. delete all nodes and edges using G.clear())
      and then add n nodes and zero edges, and return the modified graph.
  
      Thirdly, when constructing your home-grown graph creation function
      you can use empty_graph to construct the graph by passing a user
      defined create_using to empty_graph. In this case, if you want the
      default constructor to be other than nx.Graph, specify `default`.
  
      >>> def mygraph(n, create_using=None):
      ...     G = nx.empty_graph(n, create_using, nx.MultiGraph)
      ...     G.add_edges_from([(0, 1), (0, 1)])
      ...     return G
      >>> G = mygraph(3)
      >>> G.is_multigraph()
      True
      >>> G = mygraph(3, nx.Graph)
      >>> G.is_multigraph()
      False
  
      See also create_empty_copy(G).
  
      """
      if create_using is None:
          G = default()
      elif hasattr(create_using, "_adj"):
          # create_using is a NetworkX style Graph
          create_using.clear()
          G = create_using
      else:
          # try create_using as constructor
          G = create_using()
  
      n_name, nodes = n
      G.add_nodes_from(nodes)
      return G
  
  
  def ladder_graph(n, create_using=None):
      """Returns the Ladder graph of length n.
  
      This is two paths of n nodes, with
      each pair connected by a single edge.
  
      Node labels are the integers 0 to 2*n - 1.
  
      """
      G = empty_graph(2 * n, create_using)
      if G.is_directed():
          raise NetworkXError("Directed Graph not supported")
      G.add_edges_from(pairwise(range(n)))
      G.add_edges_from(pairwise(range(n, 2 * n)))
      G.add_edges_from((v, v + n) for v in range(n))
      return G
  
  
  @nodes_or_number([0, 1])
  def lollipop_graph(m, n, create_using=None):
      """Returns the Lollipop Graph; `K_m` connected to `P_n`.
  
      This is the Barbell Graph without the right barbell.
  
      Parameters
      ----------
      m, n : int or iterable container of nodes (default = 0)
          If an integer, nodes are from `range(m)` and `range(m,m+n)`.
          If a container, the entries are the coordinate of the node.
  
          The nodes for m appear in the complete graph $K_m$ and the nodes
          for n appear in the path $P_n$
      create_using : NetworkX graph constructor, optional (default=nx.Graph)
         Graph type to create. If graph instance, then cleared before populated.
  
      Notes
      -----
      The 2 subgraphs are joined via an edge (m-1, m).
      If n=0, this is merely a complete graph.
  
      (This graph is an extremal example in David Aldous and Jim
      Fill's etext on Random Walks on Graphs.)
  
      """
      m, m_nodes = m
      n, n_nodes = n
      M = len(m_nodes)
      N = len(n_nodes)
      if isinstance(m, int):
          n_nodes = [len(m_nodes) + i for i in n_nodes]
      if M < 2:
          raise NetworkXError("Invalid graph description, m should be >=2")
      if N < 0:
          raise NetworkXError("Invalid graph description, n should be >=0")
  
      # the ball
      G = complete_graph(m_nodes, create_using)
      if G.is_directed():
          raise NetworkXError("Directed Graph not supported")
      # the stick
      G.add_nodes_from(n_nodes)
      if N > 1:
          G.add_edges_from(pairwise(n_nodes))
      # connect ball to stick
      if M > 0 and N > 0:
          G.add_edge(m_nodes[-1], n_nodes[0])
      return G
  
  
  def null_graph(create_using=None):
      """Returns the Null graph with no nodes or edges.
  
      See empty_graph for the use of create_using.
  
      """
      G = empty_graph(0, create_using)
      return G
  
  
  @nodes_or_number(0)
  def path_graph(n, create_using=None):
      """Returns the Path graph `P_n` of linearly connected nodes.
  
      Parameters
      ----------
      n : int or iterable
          If an integer, nodes are 0 to n - 1.
          If an iterable of nodes, in the order they appear in the path.
      create_using : NetworkX graph constructor, optional (default=nx.Graph)
         Graph type to create. If graph instance, then cleared before populated.
  
      """
      n_name, nodes = n
      G = empty_graph(nodes, create_using)
      G.add_edges_from(pairwise(nodes))
      return G
  
  
  @nodes_or_number(0)
  def star_graph(n, create_using=None):
      """Return the star graph
  
      The star graph consists of one center node connected to n outer nodes.
  
      Parameters
      ----------
      n : int or iterable
          If an integer, node labels are 0 to n with center 0.
          If an iterable of nodes, the center is the first.
      create_using : NetworkX graph constructor, optional (default=nx.Graph)
         Graph type to create. If graph instance, then cleared before populated.
  
      Notes
      -----
      The graph has n+1 nodes for integer n.
      So star_graph(3) is the same as star_graph(range(4)).
      """
      n_name, nodes = n
      if isinstance(n_name, int):
          nodes = nodes + [n_name]  # there should be n+1 nodes
      first = nodes[0]
      G = empty_graph(nodes, create_using)
      if G.is_directed():
          raise NetworkXError("Directed Graph not supported")
      G.add_edges_from((first, v) for v in nodes[1:])
      return G
  
  
  def trivial_graph(create_using=None):
      """Return the Trivial graph with one node (with label 0) and no edges."""
      G = empty_graph(1, create_using)
      return G
  
  
  def turan_graph(n, r):
      r"""Return the Turan Graph
  
      The Turan Graph is a complete multipartite graph on $n$ nodes
      with $r$ disjoint subsets. That is, edges connect each node to
      every node not in its subset.
  
      Given $n$ and $r$, we create a complete multipartite graph with
      $r-(n \mod r)$ partitions of size $n/r$, rounded down, and
      $n \mod r$ partitions of size $n/r+1$, rounded down.
  
      Parameters
      ----------
      n : int
          The number of nodes.
      r : int
          The number of partitions.
          Must be less than or equal to n.
  
      Notes
      -----
      Must satisfy $1 <= r <= n$.
      The graph has $(r-1)(n^2)/(2r)$ edges, rounded down.
      """
  
      if not 1 <= r <= n:
          raise NetworkXError("Must satisfy 1 <= r <= n")
  
      partitions = [n // r] * (r - (n % r)) + [n // r + 1] * (n % r)
      G = complete_multipartite_graph(*partitions)
      return G
  
  
  @nodes_or_number(0)
  def wheel_graph(n, create_using=None):
      """Return the wheel graph
  
      The wheel graph consists of a hub node connected to a cycle of (n-1) nodes.
  
      Parameters
      ----------
      n : int or iterable
          If an integer, node labels are 0 to n with center 0.
          If an iterable of nodes, the center is the first.
      create_using : NetworkX graph constructor, optional (default=nx.Graph)
         Graph type to create. If graph instance, then cleared before populated.
  
      Node labels are the integers 0 to n - 1.
      """
      n_name, nodes = n
      if n_name == 0:
          G = empty_graph(0, create_using)
          return G
      G = star_graph(nodes, create_using)
      if len(G) > 2:
          G.add_edges_from(pairwise(nodes[1:]))
          G.add_edge(nodes[-1], nodes[1])
      return G
  
  
  def complete_multipartite_graph(*subset_sizes):
      """Returns the complete multipartite graph with the specified subset sizes.
  
      Parameters
      ----------
      subset_sizes : tuple of integers or tuple of node iterables
         The arguments can either all be integer number of nodes or they
         can all be iterables of nodes. If integers, they represent the
         number of nodes in each subset of the multipartite graph.
         If iterables, each is used to create the nodes for that subset.
         The length of subset_sizes is the number of subsets.
  
      Returns
      -------
      G : NetworkX Graph
         Returns the complete multipartite graph with the specified subsets.
  
         For each node, the node attribute 'subset' is an integer
         indicating which subset contains the node.
  
      Examples
      --------
      Creating a complete tripartite graph, with subsets of one, two, and three
      nodes, respectively.
  
          >>> G = nx.complete_multipartite_graph(1, 2, 3)
          >>> [G.nodes[u]["subset"] for u in G]
          [0, 1, 1, 2, 2, 2]
          >>> list(G.edges(0))
          [(0, 1), (0, 2), (0, 3), (0, 4), (0, 5)]
          >>> list(G.edges(2))
          [(2, 0), (2, 3), (2, 4), (2, 5)]
          >>> list(G.edges(4))
          [(4, 0), (4, 1), (4, 2)]
  
          >>> G = nx.complete_multipartite_graph("a", "bc", "def")
          >>> [G.nodes[u]["subset"] for u in sorted(G)]
          [0, 1, 1, 2, 2, 2]
  
      Notes
      -----
      This function generalizes several other graph builder functions.
  
      - If no subset sizes are given, this returns the null graph.
      - If a single subset size `n` is given, this returns the empty graph on
        `n` nodes.
      - If two subset sizes `m` and `n` are given, this returns the complete
        bipartite graph on `m + n` nodes.
      - If subset sizes `1` and `n` are given, this returns the star graph on
        `n + 1` nodes.
  
      See also
      --------
      complete_bipartite_graph
      """
      # The complete multipartite graph is an undirected simple graph.
      G = Graph()
  
      if len(subset_sizes) == 0:
          return G
  
      # set up subsets of nodes
      try:
          extents = pairwise(itertools.accumulate((0,) + subset_sizes))
          subsets = [range(start, end) for start, end in extents]
      except TypeError:
          subsets = subset_sizes
  
      # add nodes with subset attribute
      # while checking that ints are not mixed with iterables
      try:
          for (i, subset) in enumerate(subsets):
              G.add_nodes_from(subset, subset=i)
      except TypeError as e:
          raise NetworkXError("Arguments must be all ints or all iterables") from e
  
      # Across subsets, all nodes should be adjacent.
      # We can use itertools.combinations() because undirected.
      for subset1, subset2 in itertools.combinations(subsets, 2):
          G.add_edges_from(itertools.product(subset1, subset2))
      return G
  