wg dZddlZddlmZddgZededej d d d Zej dd d Zy)zTFunctions related to the Mycielski Operation and the Mycielskian family of graphs. N)not_implemented_for mycielskianmycielski_graphdirected multigraphT) returns_graphctj|}t|D]}|j|j tdzt |j }|jfd|D|jfd|D|jdz|jfdtD|S)a^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. c32K|]\}}||zfywN.0uvns b/home/mcse/projects/flask/flask-venv/lib/python3.12/site-packages/networkx/generators/mycielski.py zmycielskian..?s:1!QU:c32K|]\}}|z|fywr r rs rrzmycielskian..@s:1!a%:rc32K|]}|zdzfyw)r Nr )rrrs rrzmycielskian..Bs:A!a%Q:r) nxconvert_node_labels_to_integersrangenumber_of_nodesadd_nodes_fromlistedgesadd_edges_fromadd_node)G iterationsMi old_edgesrs @rrr sZ **1-A : ;    q!a%)O  : :: : :: 1q5 :q::; H)graphsrc|dkrtjd|dk(rtjdSttjd|dz S)aGenerator 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. zmust satisfy n >= 1r )r NetworkXError empty_graphr path_graph)rs rrrGsP@ 1u455Av~~a  2==+QU33r&)r)) __doc__networkxrnetworkx.utilsr__all__ _dispatchablerrr r&rr2s| . + ,Z \"%5 &#!5 pT2&43&4r&