You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
111 lines
3.2 KiB
111 lines
3.2 KiB
"""Functions related to the Mycielski Operation and the Mycielskian family
|
|
of graphs.
|
|
|
|
"""
|
|
|
|
import networkx as nx
|
|
from networkx.utils import not_implemented_for
|
|
|
|
__all__ = ["mycielskian", "mycielski_graph"]
|
|
|
|
|
|
@not_implemented_for("directed")
|
|
@not_implemented_for("multigraph")
|
|
@nx._dispatchable(returns_graph=True)
|
|
def mycielskian(G, iterations=1):
|
|
r"""Returns the Mycielskian of a simple, undirected graph G
|
|
|
|
The Mycielskian of graph preserves a graph's triangle free
|
|
property while increasing the chromatic number by 1.
|
|
|
|
The Mycielski Operation on a graph, :math:`G=(V, E)`, constructs a new
|
|
graph with :math:`2|V| + 1` nodes and :math:`3|E| + |V|` edges.
|
|
|
|
The construction is as follows:
|
|
|
|
Let :math:`V = {0, ..., n-1}`. Construct another vertex set
|
|
:math:`U = {n, ..., 2n}` and a vertex, `w`.
|
|
Construct a new graph, `M`, with vertices :math:`U \bigcup V \bigcup w`.
|
|
For edges, :math:`(u, v) \in E` add edges :math:`(u, v), (u, v + n)`, and
|
|
:math:`(u + n, v)` to M. Finally, for all vertices :math:`u \in U`, add
|
|
edge :math:`(u, w)` to M.
|
|
|
|
The Mycielski Operation can be done multiple times by repeating the above
|
|
process iteratively.
|
|
|
|
More information can be found at https://en.wikipedia.org/wiki/Mycielskian
|
|
|
|
Parameters
|
|
----------
|
|
G : graph
|
|
A simple, undirected NetworkX graph
|
|
iterations : int
|
|
The number of iterations of the Mycielski operation to
|
|
perform on G. Defaults to 1. Must be a non-negative integer.
|
|
|
|
Returns
|
|
-------
|
|
M : graph
|
|
The Mycielskian of G after the specified number of iterations.
|
|
|
|
Notes
|
|
-----
|
|
Graph, node, and edge data are not necessarily propagated to the new graph.
|
|
|
|
"""
|
|
|
|
M = nx.convert_node_labels_to_integers(G)
|
|
|
|
for i in range(iterations):
|
|
n = M.number_of_nodes()
|
|
M.add_nodes_from(range(n, 2 * n))
|
|
old_edges = list(M.edges())
|
|
M.add_edges_from((u, v + n) for u, v in old_edges)
|
|
M.add_edges_from((u + n, v) for u, v in old_edges)
|
|
M.add_node(2 * n)
|
|
M.add_edges_from((u + n, 2 * n) for u in range(n))
|
|
|
|
return M
|
|
|
|
|
|
@nx._dispatchable(graphs=None, returns_graph=True)
|
|
def mycielski_graph(n):
|
|
"""Generator for the n_th Mycielski Graph.
|
|
|
|
The Mycielski family of graphs is an infinite set of graphs.
|
|
:math:`M_1` is the singleton graph, :math:`M_2` is two vertices with an
|
|
edge, and, for :math:`i > 2`, :math:`M_i` is the Mycielskian of
|
|
:math:`M_{i-1}`.
|
|
|
|
More information can be found at
|
|
http://mathworld.wolfram.com/MycielskiGraph.html
|
|
|
|
Parameters
|
|
----------
|
|
n : int
|
|
The desired Mycielski Graph.
|
|
|
|
Returns
|
|
-------
|
|
M : graph
|
|
The n_th Mycielski Graph
|
|
|
|
Notes
|
|
-----
|
|
The first graph in the Mycielski sequence is the singleton graph.
|
|
The Mycielskian of this graph is not the :math:`P_2` graph, but rather the
|
|
:math:`P_2` graph with an extra, isolated vertex. The second Mycielski
|
|
graph is the :math:`P_2` graph, so the first two are hard coded.
|
|
The remaining graphs are generated using the Mycielski operation.
|
|
|
|
"""
|
|
|
|
if n < 1:
|
|
raise nx.NetworkXError("must satisfy n >= 1")
|
|
|
|
if n == 1:
|
|
return nx.empty_graph(1)
|
|
|
|
else:
|
|
return mycielskian(nx.path_graph(2), n - 2)
|