def annihilation_number(G):
r"""Return the annihilation number of the graph.
The annihilation number of a graph G is defined as:
.. math::
a(G) = \max\{t : \sum_{i=0}^t d_i \leq m\}
where
.. math::
{d_1 \leq d_2 \leq \cdots \leq d_n}
is the degree sequence of the graph ordered in non-decreasing order and *m*
is the number of edges in G.
Parameters
----------
G : NetworkX graph
An undirected graph.
Returns
-------
int
The annihilation number of the graph.
"""
D = degree_sequence(G)
D.sort() # sort in non-decreasing order
n = len(D)
m = number_of_edges(G)
# sum over degrees in the sequence until the sum is larger than the number of edges in the graph
for i in reversed(range(n + 1)):
if sum(D[:i]) <= m:
return i
def annihilation_number(G):
# TODO: Add documentation
D = degree_sequence(G)
D.sort() # sort in non-decreasing order
m = number_of_edges(G)
# sum over degrees in the sequence until the sum is larger than the number of edges in the graph
S = [D[0]]
while(sum(S) <= m):
S.append(D[len(S)])
return len(S) - 1
def max_matching_bf(G):
"""Return a maximum matching in G.
A *maximum matching* is a largest set of edges such that no two
edges in the set have a common endpoint.
Parameters
----------
G : NetworkX graph
An undirected graph.
Returns
-------
set
A set of edges in a maximum matching.
"""
if number_of_edges(G) == 0:
return set()
for i in reversed(range(1, number_of_edges(G) + 1)):
for S in combinations(edges(G), i):
if is_matching(G, set(S)):
return set(S)
def max_k_independent_set(G, k):
# TODO: Add documentation
# set the maximum for the loop range
rangeMax = number_of_nodes(G) + 1
if k == 1:
rangeMax = annihilation_number(G) + 1
elif number_of_edges(G) > 0:
rangeMax = number_of_nodes(G)
# TODO: can the above range be improved with some general upper bound for the k-independence number?
# loop through subsets of nodes of G in decreasing order of size until a k-independent set is found
for i in reversed(range(rangeMax)):
for S in combinations(nodes(G), i):
if is_k_independent_set(G, S, k):
return list(S)
# return None if no independent set is found (should not occur)
return None
def test_matching_number_of_path_is_ceil_of_half_of_edges():
for i in range(2, 12):
P = gp.path_graph(i)
m = math.ceil(gp.number_of_edges(P) / 2)
assert gp.matching_number(P, method="bf") == m
assert gp.matching_number(P, method="ilp") == m