def CompleteBipartiteGraph(p, q, set_position=True):
r"""
Return a Complete Bipartite Graph on `p + q` vertices.
A Complete Bipartite Graph is a graph with its vertices partitioned into two
groups, `V_1 = \{0,...,p-1\}` and `V_2 = \{p,...,p+q-1\}`. Each `u \in
V_1` is connected to every `v \in V_2`.
INPUT:
- ``p,q`` -- number of vertices in each side
- ``set_position`` -- boolean (default ``True``); if set to ``True``, we
assign positions to the vertices so that the set of cardinality `p` is
on the line `y=1` and the set of cardinality `q` is on the line `y=0`.
PLOTTING: Upon construction, the position dictionary is filled to override
the spring-layout algorithm. By convention, each complete bipartite graph
will be displayed with the first `p` nodes on the top row (at `y=1`) from
left to right. The remaining `q` nodes appear at `y=0`, also from left to
right. The shorter row (partition with fewer nodes) is stretched to the same
length as the longer row, unless the shorter row has 1 node; in which case
it is centered. The `x` values in the plot are in domain `[0, \max(p, q)]`.
In the Complete Bipartite graph, there is a visual difference in using the
spring-layout algorithm vs. the position dictionary used in this
constructor. The position dictionary flattens the graph and separates the
partitioned nodes, making it clear which nodes an edge is connected to. The
Complete Bipartite graph plotted with the spring-layout algorithm tends to
center the nodes in `p` (see ``spring_med`` in examples below), thus
overlapping its nodes and edges, making it typically hard to decipher.
Filling the position dictionary in advance adds `O(n)` to the constructor.
Feel free to race the constructors below in the examples section. The much
larger difference is the time added by the spring-layout algorithm when
plotting. (Also shown in the example below). The spring model is typically
described as `O(n^3)`, as appears to be the case in the NetworkX source
code.
EXAMPLES:
Two ways of constructing the complete bipartite graph, using different
layout algorithms::
sage: import networkx
sage: n = networkx.complete_bipartite_graph(389, 157); spring_big = Graph(n) # long time
sage: posdict_big = graphs.CompleteBipartiteGraph(389, 157) # long time
Compare the plotting::
sage: n = networkx.complete_bipartite_graph(11, 17)
sage: spring_med = Graph(n)
sage: posdict_med = graphs.CompleteBipartiteGraph(11, 17)
Notice here how the spring-layout tends to center the nodes of `n1`::
sage: spring_med.show() # long time
sage: posdict_med.show() # long time
View many complete bipartite graphs with a Sage Graphics Array, with this
constructor (i.e., the position dictionary filled)::
sage: g = []
sage: j = []
sage: for i in range(9):
....: k = graphs.CompleteBipartiteGraph(i+1,4)
....: 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 = graphics_array(j)
sage: G.show() # long time
We compare to plotting with the spring-layout algorithm::
sage: g = []
sage: j = []
sage: for i in range(9):
....: spr = networkx.complete_bipartite_graph(i+1,4)
....: k = Graph(spr)
....: 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 = graphics_array(j)
sage: G.show() # long time
:trac:`12155`::
sage: graphs.CompleteBipartiteGraph(5,6).complement()
complement(Complete bipartite graph of order 5+6): Graph on 11 vertices
TESTS:
Prevent negative dimensions (:trac:`18530`)::
sage: graphs.CompleteBipartiteGraph(-1,1)
Traceback (most recent call last):
...
ValueError: the arguments p(=-1) and q(=1) must be positive integers
sage: graphs.CompleteBipartiteGraph(1,-1)
Traceback (most recent call last):
...
ValueError: the arguments p(=1) and q(=-1) must be positive integers
"""
if p < 0 or q < 0:
raise ValueError('the arguments p(={}) and q(={}) must be positive integers'.format(p, q))
G = Graph(p + q, name="Complete bipartite graph of order {}+{}".format(p, q))
G.add_edges((i, j) for i in range(p) for j in range(p, p + q))
# We now assign positions to vertices:
# - vertices 0,..,p-1 are placed on the line (0, 1) to (max(p, q), 1)
# - vertices p,..,p+q-1 are placed on the line (0, 0) to (max(p, q), 0)
# If p (or q) is 1, the vertex is centered in the line.
if set_position:
nmax = max(p, q)
G._line_embedding(list(range(p)), first=(0, 1), last=(nmax, 1))
G._line_embedding(list(range(p, p + q)), first=(0, 0), last=(nmax, 0))
return G
def CompleteMultipartiteGraph(l):
r"""
Return a complete multipartite graph.
INPUT:
- ``l`` -- a list of integers; the respective sizes of the components
PLOTTING: Produce a layout of the vertices so that vertices in the same
vertex set are adjacent and clearly separated from vertices in other vertex
sets.
This is done by calculating the vertices of an `r`-gon then calculating the
slope between adjacent vertices. We then 'walk' around the `r`-gon placing
graph vertices in regular intervals between adjacent vertices of the
`r`-gon.
Makes a nicely organized graph like in this picture:
https://commons.wikimedia.org/wiki/File:Turan_13-4.svg
EXAMPLES:
A complete tripartite graph with sets of sizes `5, 6, 8`::
sage: g = graphs.CompleteMultipartiteGraph([5, 6, 8]); g
Multipartite Graph with set sizes [5, 6, 8]: Graph on 19 vertices
It clearly has a chromatic number of 3::
sage: g.chromatic_number()
3
"""
r = len(l) # getting the number of partitions
name = "Multipartite Graph with set sizes {}".format(l)
if not r:
g = Graph()
elif r == 1:
g = Graph(l[0])
g._line_embedding(range(l[0]), first=(0, 0), last=(l[0], 0))
elif r == 2:
g = CompleteBipartiteGraph(l[0], l[1])
g.name(name)
else:
# This position code gives bad results on bipartite or isolated graphs
points = [(cos(2 * pi * i / r), sin(2 * pi * i / r)) for i in range(r)]
slopes = [(points[(i + 1) % r][0] - points[i % r][0],
points[(i + 1) % r][1] - points[i % r][1]) for i in range(r)]
counter = 0
positions = {}
for i in range(r):
vertex_set_size = l[i] + 1
for j in range(1, vertex_set_size):
x = points[i][0] + slopes[i][0] * j / vertex_set_size
y = points[i][1] + slopes[i][1] * j / vertex_set_size
positions[counter] = (x, y)
counter += 1
g = Graph(sum(l))
s = 0
for i in l:
g.add_clique(range(s, s + i))
s += i
g = g.complement()
g.set_pos(positions)
g.name(name)
return g
def CompleteBipartiteGraph(n1, n2, set_position=True):
r"""
Return a Complete Bipartite Graph on `n1 + n2` vertices.
A Complete Bipartite Graph is a graph with its vertices partitioned into two
groups, `V_1 = \{0,...,n1-1\}` and `V_2 = \{n1,...,n1+n2-1\}`. Each `u \in
V_1` is connected to every `v \in V_2`.
INPUT:
- ``n1, n2`` -- number of vertices in each side
- ``set_position`` -- boolean (default ``True``); if set to ``True``, we
assign positions to the vertices so that the set of cardinality `n1` is
on the line `y=1` and the set of cardinality `n2` is on the line `y=0`.
PLOTTING: Upon construction, the position dictionary is filled to override
the spring-layout algorithm. By convention, each complete bipartite graph
will be displayed with the first `n1` nodes on the top row (at `y=1`) from
left to right. The remaining `n2` nodes appear at `y=0`, also from left to
right. The shorter row (partition with fewer nodes) is stretched to the same
length as the longer row, unless the shorter row has 1 node; in which case
it is centered. The `x` values in the plot are in domain `[0, \max(n1,
n2)]`.
In the Complete Bipartite graph, there is a visual difference in using the
spring-layout algorithm vs. the position dictionary used in this
constructor. The position dictionary flattens the graph and separates the
partitioned nodes, making it clear which nodes an edge is connected to. The
Complete Bipartite graph plotted with the spring-layout algorithm tends to
center the nodes in n1 (see spring_med in examples below), thus overlapping
its nodes and edges, making it typically hard to decipher.
Filling the position dictionary in advance adds `O(n)` to the constructor.
Feel free to race the constructors below in the examples section. The much
larger difference is the time added by the spring-layout algorithm when
plotting. (Also shown in the example below). The spring model is typically
described as `O(n^3)`, as appears to be the case in the NetworkX source
code.
EXAMPLES:
Two ways of constructing the complete bipartite graph, using different
layout algorithms::
sage: import networkx
sage: n = networkx.complete_bipartite_graph(389, 157); spring_big = Graph(n) # long time
sage: posdict_big = graphs.CompleteBipartiteGraph(389, 157) # long time
Compare the plotting::
sage: n = networkx.complete_bipartite_graph(11, 17)
sage: spring_med = Graph(n)
sage: posdict_med = graphs.CompleteBipartiteGraph(11, 17)
Notice here how the spring-layout tends to center the nodes of `n1`::
sage: spring_med.show() # long time
sage: posdict_med.show() # long time
View many complete bipartite graphs with a Sage Graphics Array, with this
constructor (i.e., the position dictionary filled)::
sage: g = []
sage: j = []
sage: for i in range(9):
....: k = graphs.CompleteBipartiteGraph(i+1,4)
....: 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
We compare to plotting with the spring-layout algorithm::
sage: g = []
sage: j = []
sage: for i in range(9):
....: spr = networkx.complete_bipartite_graph(i+1,4)
....: k = Graph(spr)
....: 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
:trac:`12155`::
sage: graphs.CompleteBipartiteGraph(5,6).complement()
complement(Complete bipartite graph of order 5+6): Graph on 11 vertices
TESTS:
Prevent negative dimensions (:trac:`18530`)::
sage: graphs.CompleteBipartiteGraph(-1,1)
Traceback (most recent call last):
...
ValueError: the arguments n1(=-1) and n2(=1) must be positive integers
sage: graphs.CompleteBipartiteGraph(1,-1)
Traceback (most recent call last):
...
ValueError: the arguments n1(=1) and n2(=-1) must be positive integers
"""
if n1<0 or n2<0:
raise ValueError('the arguments n1(={}) and n2(={}) must be positive integers'.format(n1,n2))
G = Graph(n1+n2, name="Complete bipartite graph of order {}+{}".format(n1, n2))
G.add_edges((i,j) for i in range(n1) for j in range(n1,n1+n2))
# We now assign positions to vertices:
# - vertices 0,..,n1-1 are placed on the line (0, 1) to (max(n1, n2), 1)
# - vertices n1,..,n1+n2-1 are placed on the line (0, 0) to (max(n1, n2), 0)
# If n1 (or n2) is 1, the vertex is centered in the line.
if set_position:
nmax = max(n1, n2)
G._line_embedding(list(range(n1)), first=(0, 1), last=(nmax, 1))
G._line_embedding(list(range(n1, n1+n2)), first=(0, 0), last=(nmax, 0))
return G