def min_connected_k_forcing_set(G, k):
"""Return a smallest connected k-forcing set in *G*.
The method used to compute the set is brute force.
Parameters
----------
G : NetworkX graph
An undirected graph.
k : int
A positive integer
Returns
-------
list
A list of nodes in a smallest connected k-forcing set in *G*.
"""
# check that k is a positive integer
if not float(k).is_integer():
raise TypeError("Expected k to be an integer.")
k = int(k)
if k < 1:
raise ValueError("Expected k to be a positive integer.")
# only start search if graph is connected
if not is_connected(G):
return None
for i in range(1, G.order() + 1):
for S in combinations(G.nodes(), i):
if is_connected_k_forcing_set(G, S, k):
return list(S)
def is_connected_k_forcing_set(G, nodes, k):
"""Return whether or not the nodes in `nodes` comprise a connected k-forcing
set in *G*.
A *connected k-forcing set* in a graph *G* is a k-forcing set that induces
a connected subgraph.
Parameters
----------
G : NetworkX graph
An undirected graph.
nodes : list, set
An iterable container of nodes in G.
k : int
A positive integer.
Returns
-------
boolean
True if the nodes in nbunch comprise a connected k-forcing set in *G*.
False otherwise.
"""
# check that k is a positive integer
if not float(k).is_integer():
raise TypeError("Expected k to be an integer.")
k = int(k)
if k < 1:
raise ValueError("Expected k to be a positive integer.")
S = set(n for n in nodes if n in G)
H = G.subgraph(S)
return is_connected(H) and is_k_forcing_set(G, S, k)
def is_connected_k_dominating_set(G, nodes, k):
"""Return whether or not *nodes* is a connected *k*-dominating set of *G*.
A set *D* is a *connected k-dominating set* of *G* is *D* is a
*k*-dominating set in *G* and the subgraph of *G* induced by *D* is
a connected graph.
Parameters
----------
G : NetworkX graph
An undirected graph.
nodes : list, set
An iterable container of nodes in G.
k : int
A positive integer
Returns
-------
boolean
True if *nbunch* is a connected *k*-dominating set in *G*, and false
otherwise.
"""
# check that k is a positive integer
if not float(k).is_integer():
raise TypeError("Expected k to be an integer.")
k = int(k)
if k < 1:
raise ValueError("Expected k to be a positive integer.")
S = set(n for n in nodes if n in G)
H = G.subgraph(S)
return is_connected(H) and is_k_dominating_set(G, S, k)
def min_connected_k_dominating_set(G, k):
"""Return a smallest connected k-dominating set in the graph.
The method to compute the set is brute force.
Parameters
----------
G : NetworkX graph
An undirected graph.
k : int
A positive integer.
Returns
-------
list
A list of nodes in a smallest k-dominating set in the graph.
"""
# check that k is a positive integer
if not float(k).is_integer():
raise TypeError("Expected k to be an integer.")
k = int(k)
if k < 1:
raise ValueError("Expected k to be a positive integer.")
# Only proceed with search if graph is connected
if not is_connected(G):
return []
for i in range(1, number_of_nodes(G) + 1):
for S in combinations(nodes(G), i):
if is_connected_k_dominating_set(G, S, k):
return list(S)
def chromatic_number_ram_rama(G):
"""Return the chromatic number of G.
The *chromatic number* of a graph G is the size of a mininum coloring of
the nodes in G such that no two adjacent nodes have the same color.
The method for computing the chromatic number is an implementation of the
algorithm discovered by Ram and Rama.
Parameters
----------
G : NetworkX graph
An undirected graph.
Returns
-------
int
The chromatic number of G.
References
----------
A.M. Ram, R. Rama, An alternate method to find the chromatic number of a
finite, connected graph, *arXiv preprint
arXiv:1309.3642*, (2013)
"""
if not is_connected(G):
raise TypeError("Invalid graph: not connected")
if is_complete_graph(G):
return G.order()
# get list of pairs of non neighbors in G
N = [
list(p) for p in pairs_of_nodes(G) if not are_neighbors(G, p[0], p[1])
]
# get a pair of non neighbors who have the most common neighbors
num_common_neighbors = list(map(lambda p: len(common_neighbors(G, p)), N))
P = N[np.argmax(num_common_neighbors)]
# Collapse the nodes in P and repeat the above process
H = G.copy()
contract_nodes(H, P)
return chromatic_number(H)