Source code for networkx.generators.mycielski

"""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") @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 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
[docs] @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 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)