def __init__(self, ID, order, adj_matrix, line_number):
self.ID = ID
#TODO Fix me for string input
self.order = order
self.internal_graph = CompleteGraph(self.order)
self.adj_matrix = adj_matrix
self.line_number = line_number
def __init__(self, ID, order, adj_matrix, line_number):
self.ID = ID
# TODO Fix me for string input
self.order = order
self.internal_graph = CompleteGraph(self.order)
self.adj_matrix = adj_matrix
self.line_number = line_number
def is_transversal_design(B, k, n, verbose=False):
r"""
Check that a given set of blocks ``B`` is a transversal design.
See :func:`~sage.combinat.designs.orthogonal_arrays.transversal_design`
for a definition.
INPUT:
- ``B`` -- the list of blocks
- ``k, n`` -- integers
- ``verbose`` (boolean) -- whether to display information about what is
going wrong.
.. NOTE::
The tranversal design must have `\{0, \ldots, kn-1\}` as a ground set,
partitioned as `k` sets of size `n`: `\{0, \ldots, k-1\} \sqcup
\{k, \ldots, 2k-1\} \sqcup \cdots \sqcup \{k(n-1), \ldots, kn-1\}`.
EXAMPLES::
sage: TD = designs.transversal_design(5, 5, check=True) # indirect doctest
sage: from sage.combinat.designs.orthogonal_arrays import is_transversal_design
sage: is_transversal_design(TD, 5, 5)
True
sage: is_transversal_design(TD, 4, 4)
False
"""
from sage.graphs.generators.basic import CompleteGraph
from itertools import combinations
g = k * CompleteGraph(n)
m = g.size()
for X in B:
if len(X) != k:
if verbose:
print "A set has wrong size"
return False
g.add_edges(list(combinations(X, 2)))
if g.size() != m + (len(X) * (len(X) - 1)) / 2:
if verbose:
print "A pair appears twice"
return False
m = g.size()
if not g.is_clique():
if verbose:
print "A pair did not appear"
return False
return True
class BaseGraph:
def __init__(self, ID, order, adj_matrix, line_number):
self.ID = ID
# TODO Fix me for string input
self.order = order
self.internal_graph = CompleteGraph(self.order)
self.adj_matrix = adj_matrix
self.line_number = line_number
def instantiate(self, solver, options):
prev_vars = solver.nvars()
solver.add_vars(self, self.internal_graph.vertices())
solver.add_vars(self, self.internal_graph.edges(labels=False))
solver.add_instantiate_graph_constraint(edge_to_vertex_axiom(self, solver))
curr_vars = solver.nvars()
solver.graph_vars.append((self, prev_vars + 1, curr_vars))
def create_graph_from_model(self, solver, model):
"""
Given a model from the SAT solver, construct the graph.
"""
g = Graph()
on_vertices = solver.get_objects_in_model(model, self, self.internal_graph.vertices())
on_edges = solver.get_objects_in_model(model, self, self.internal_graph.edges(labels=False))
g.add_vertices(on_vertices)
g.add_edges(on_edges)
return g
def vertices(self):
return self.internal_graph.vertices()
def edges(self):
return self.internal_graph.edges()
def __str__(self):
return (
"graph "
+ self.ID.ID
+ "("
+ str(self.order)
+ (":" + str(self.adj_matrix) if self.adj_matrix else "")
+ ")"
)
def __repr__(self):
return self.__str__()
def toStr(self, indent):
return common.INDENT * indent + self.__str__()
class BaseGraph():
def __init__(self, ID, order, adj_matrix, line_number):
self.ID = ID
#TODO Fix me for string input
self.order = order
self.internal_graph = CompleteGraph(self.order)
self.adj_matrix = adj_matrix
self.line_number = line_number
def instantiate(self, solver, options):
prev_vars = solver.nvars()
solver.add_vars(self, self.internal_graph.vertices())
solver.add_vars(self, self.internal_graph.edges(labels=False))
solver.add_instantiate_graph_constraint(
edge_to_vertex_axiom(self, solver))
curr_vars = solver.nvars()
solver.graph_vars.append((self, prev_vars + 1, curr_vars))
def create_graph_from_model(self, solver, model):
'''
Given a model from the SAT solver, construct the graph.
'''
g = Graph()
on_vertices = solver.get_objects_in_model(
model, self, self.internal_graph.vertices())
on_edges = solver.get_objects_in_model(
model, self, self.internal_graph.edges(labels=False))
g.add_vertices(on_vertices)
g.add_edges(on_edges)
return g
def vertices(self):
return self.internal_graph.vertices()
def edges(self):
return self.internal_graph.edges()
def __str__(self):
return "graph " + self.ID.ID + "(" + str(self.order) + (
":" + str(self.adj_matrix) if self.adj_matrix else "") + ")"
def __repr__(self):
return self.__str__()
def toStr(self, indent):
return common.INDENT * indent + self.__str__()
def RandomGNP(n, p, seed=None, fast=True, method='Sage'):
r"""
Returns a random graph on `n` nodes. Each edge is inserted independently
with probability `p`.
INPUTS:
- ``n`` -- number of nodes of the digraph
- ``p`` -- probability of an edge
- ``seed`` -- integer seed for random number generator (default=None).
- ``fast`` -- boolean set to True (default) to use the algorithm with
time complexity in `O(n+m)` proposed in [3]_. It is designed for
generating large sparse graphs. It is faster than other methods for
*LARGE* instances (try it to know whether it is useful for you).
- ``method`` -- By default (```method='Sage'``), this function uses the
method implemented in ```sage.graphs.graph_generators_pyx.pyx``. When
``method='networkx'``, this function calls the NetworkX function
``fast_gnp_random_graph``, unless ``fast=False``, then
``gnp_random_graph``. Try them to know which method is the best for
you. The ``fast`` parameter is not taken into account by the 'Sage'
method so far.
REFERENCES:
.. [1] P. Erdos and A. Renyi. On Random Graphs, Publ. Math. 6, 290 (1959).
.. [2] E. N. Gilbert. Random Graphs, Ann. Math. Stat., 30, 1141 (1959).
.. [3] V. Batagelj and U. Brandes. Efficient generation of large
random networks. Phys. Rev. E, 71, 036113, 2005.
PLOTTING: When plotting, this graph will use the default spring-layout
algorithm, unless a position dictionary is specified.
EXAMPLES: We show the edge list of a random graph on 6 nodes with
probability `p = .4`::
sage: set_random_seed(0)
sage: graphs.RandomGNP(6, .4).edges(labels=False)
[(0, 1), (0, 5), (1, 2), (2, 4), (3, 4), (3, 5), (4, 5)]
We plot a random graph on 12 nodes with probability
`p = .71`::
sage: gnp = graphs.RandomGNP(12,.71)
sage: gnp.show() # long time
We view many random graphs using a graphics array::
sage: g = []
sage: j = []
sage: for i in range(9):
... k = graphs.RandomGNP(i+3,.43)
... g.append(k)
...
sage: for i in range(3):
... n = []
... for m in range(3):
... n.append(g[3*i + m].plot(vertex_size=50, vertex_labels=False))
... j.append(n)
...
sage: G = sage.plot.graphics.GraphicsArray(j)
sage: G.show() # long time
sage: graphs.RandomGNP(4,1)
Complete graph: Graph on 4 vertices
TESTS::
sage: graphs.RandomGNP(50,.2,method=50)
Traceback (most recent call last):
...
ValueError: 'method' must be equal to 'networkx' or to 'Sage'.
sage: set_random_seed(0)
sage: graphs.RandomGNP(50,.2, method="Sage").size()
243
sage: graphs.RandomGNP(50,.2, method="networkx").size()
258
"""
if n < 0:
raise ValueError("The number of nodes must be positive or null.")
if 0.0 > p or 1.0 < p:
raise ValueError("The probability p must be in [0..1].")
if seed is None:
seed = current_randstate().long_seed()
if p == 1:
from sage.graphs.generators.basic import CompleteGraph
return CompleteGraph(n)
if method == 'networkx':
import networkx
if fast:
G = networkx.fast_gnp_random_graph(n, p, seed=seed)
else:
G = networkx.gnp_random_graph(n, p, seed=seed)
return graph.Graph(G)
elif method in ['Sage', 'sage']:
# We use the Sage generator
from sage.graphs.graph_generators_pyx import RandomGNP as sageGNP
return sageGNP(n, p)
else:
raise ValueError("'method' must be equal to 'networkx' or to 'Sage'.")
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),
certify=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),
certify=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),
certify=True)[1])
# From now on we can assume (thanks to Beineke) that no edge belongs to two
# even triangles at once.
error_message = ("It looks like there is a problem somewhere. You"
"found a bug here ! Please report it on sage-devel,"
"our google group !")
# Better to work on integers... Everything takes more time
# otherwise.
G = g.relabel(inplace=False)
# Dictionary of (pairs of) cliques, i.e. the two cliques
# associated with each vertex.
v_cliques = {v: [] for v in G}
# All the even triangles we meet
even_triangles = []
# Here is THE "problem" of this implementation. Listing all maximal cliques
# takes an exponential time on general graphs (while it is obviously
# polynomial on line graphs). The problem is that this implementation cannot
# be used to *recognise* line graphs for as long as cliques_maximal returns
# a list and does not ITERATE on the maximal cliques : if there are too many
# cliques in the graph, this implementation will notice it and answer that
# the graph is not a line graph. If, on the other hand, the first thing it
# does is enumerate ALL maximal cliques, then there is no way to say early
# that the graph is not a line graph.
#
# If this cliques_maximal thing is replaced by an iterator that does not
# build the list of all cliques before returning them, then this method is a
# good recognition algorithm.
for S in G.cliques_maximal():
# Triangles... even or odd ?
if len(S) == 3:
# If a vertex of G has an odd number of neighbors among the vertices
# of S, then the triangle is odd. We compute the list of such
# vertices by taking the symmetric difference of the neighborhood of
# our three vertices.
#
# Note that the elements of S do not appear in this set as they are
# all seen exactly twice.
odd_neighbors = set(G.neighbors(S[0]))
odd_neighbors.symmetric_difference_update(G.neighbors(S[1]))
odd_neighbors.symmetric_difference_update(G.neighbors(S[2]))
# Even triangles
if not odd_neighbors:
even_triangles.append(tuple(S))
continue
# We manage odd triangles the same way we manage other cliques ...
# We now associate the clique to all the vertices it contains.
for v in S:
if len(v_cliques[v]) == 2:
raise ValueError("This graph is not a line graph !")
v_cliques[v].append(tuple(S))
if verbose:
print("Added clique", S)
# Deal with even triangles
for u, v, w in even_triangles:
# According to Beineke, we must go through all even triangles, and for
# each triangle uvw consider its three pairs of adjacent verties uv, vw,
# wu. For all pairs xy among those such that xy do not appear together
# in any clique we have found so far, we add xy to the list of cliques
# describing our covering.
for x, y in [(u, v), (v, w), (w, u)]:
# If edge xy does not appear in any of the cliques associated with y
if all([not x in C for C in v_cliques[y]]):
if len(v_cliques[y]) >= 2 or len(v_cliques[x]) >= 2:
raise ValueError("This graph is not a line graph !")
v_cliques[x].append((x, y))
v_cliques[y].append((x, y))
if verbose:
print("Adding pair", (x, y),
"appearing in the even triangle", (u, v, w))
# Deal with vertices contained in only one clique. All edges must be defined
# by TWO endpoints, so we add a fake clique.
for x, clique_list in v_cliques.iteritems():
if len(clique_list) == 1:
clique_list.append((x, ))
# We now have all our cliques. Let's build the root graph to check that it
# all fits !
from sage.graphs.graph import Graph
R = Graph()
# Associates an integer to each clique
relabel = {}
# Associates to each vertex of G its pair of coordinates in R
vertex_to_map = {}
for v, L in v_cliques.iteritems():
# Add cliques to relabel dictionary
for S in L:
if not S in relabel:
relabel[S] = len(relabel)
# The coordinates of edge v
vertex_to_map[v] = relabel[L[0]], relabel[L[1]]
if verbose:
print("Final associations :")
for v, L in v_cliques.iteritems():
print(v, L)
# We now build R
R.add_edges(vertex_to_map.values())
# Even if whatever is written above is complete nonsense, here we
# make sure that we do not return gibberish. Is the line graph of
# R isomorphic to the input ? If so, we return R, and the
# isomorphism. Else, we panic and scream.
#
# It's actually "just to make sure twice". This can be removed later if it
# turns out to be too costly.
is_isom, isom = g.is_isomorphic(R.line_graph(labels=False), certify=True)
if not is_isom:
raise Exception(error_message)
return R, isom