def bipartite_graph(self):
r"""
Construct a ``BipartiteGraph`` Object of the game.
This method is similar to the plot method.
Note that the game must be solved for this to work.
EXAMPLES:
An error is returned if the game is not solved::
sage: suit = {0: (3, 4),
....: 1: (3, 4)}
sage: revr = {3: (0, 1),
....: 4: (1, 0)}
sage: g = MatchingGame([suit, revr])
sage: g.bipartite_graph()
Traceback (most recent call last):
...
ValueError: game has not been solved yet
sage: g.solve()
{0: 3, 1: 4}
sage: g.bipartite_graph()
Bipartite graph on 4 vertices
"""
self._is_solved()
graph = BipartiteGraph(self._sol_dict)
return graph
def incidence_graph(self):
"""
Returns the incidence graph of the design, where the incidence
matrix of the design is the adjacency matrix of the graph.
EXAMPLE::
sage: BD = IncidenceStructure(range(7),[[0,1,2],[0,3,4],[0,5,6],[1,3,5],[1,4,6],[2,3,6],[2,4,5]])
sage: BD.incidence_graph()
Bipartite graph on 14 vertices
sage: A = BD.incidence_matrix()
sage: Graph(block_matrix([[A*0,A],[A.transpose(),A*0]])) == BD.incidence_graph()
True
REFERENCE:
- Sage Reference Manual on Graphs
"""
from sage.graphs.bipartite_graph import BipartiteGraph
A = self.incidence_matrix()
return BipartiteGraph(A)
def DegreeSequenceBipartite(s1, s2):
r"""
Returns a bipartite graph whose two sets have the given
degree sequences.
Given two different sequences of degrees `s_1` and `s_2`,
this functions returns ( if possible ) a bipartite graph
on sets `A` and `B` such that the vertices in `A` have
`s_1` as their degree sequence, while `s_2` is the degree
sequence of the vertices in `B`.
INPUT:
- ``s_1`` -- list of integers corresponding to the degree
sequence of the first set.
- ``s_2`` -- list of integers corresponding to the degree
sequence of the second set.
ALGORITHM:
This function works through the computation of the matrix
given by the Gale-Ryser theorem, which is in this case
the adjacency matrix of the bipartite graph.
EXAMPLES:
If we are given as sequences ``[2,2,2,2,2]`` and ``[5,5]``
we are given as expected the complete bipartite
graph `K_{2,5}` ::
sage: g = graphs.DegreeSequenceBipartite([2,2,2,2,2],[5,5])
sage: g.is_isomorphic(graphs.CompleteBipartiteGraph(5,2))
True
Some sequences being incompatible if, for example, their sums
are different, the functions raises a ``ValueError`` when no
graph corresponding to the degree sequences exists. ::
sage: g = graphs.DegreeSequenceBipartite([2,2,2,2,1],[5,5])
Traceback (most recent call last):
...
ValueError: There exists no bipartite graph corresponding to the given degree sequences
TESTS:
:trac:`12155`::
sage: graphs.DegreeSequenceBipartite([2,2,2,2,2],[5,5]).complement()
Graph on 7 vertices
"""
from sage.combinat.integer_vector import gale_ryser_theorem
from sage.graphs.bipartite_graph import BipartiteGraph
s1 = sorted(s1, reverse=True)
s2 = sorted(s2, reverse=True)
m = gale_ryser_theorem(s1, s2)
if m is False:
raise ValueError(
"There exists no bipartite graph corresponding to the given degree sequences"
)
else:
return Graph(BipartiteGraph(m))