"""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"]
[docs]@not_implemented_for("directed")
@not_implemented_for("multigraph")
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.
"""
n = G.number_of_nodes()
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
[docs]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 >= 0")
if n == 1:
return nx.empty_graph(1)
else:
return mycielskian(nx.path_graph(2), n - 2)