\Mh dZddlZddlmZddgZededejd dd Zejdd d ZdS)zTFunctions related to the Mycielski Operation and the Mycielskian family of graphs. N)not_implemented_for mycielskianmycielski_graphdirected multigraphT) returns_graphctj|}t|D]}||tdzt |}|fd|D|fd|D|dz|fdtD|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. c3,K|]\}}||zfVdSN.0uvns ]/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/networkx/generators/mycielski.py zmycielskian..?s/::1!QU::::::c3,K|]\}}|z|fVdSr rrs rrzmycielskian..@s/::1!a%::::::rc3,K|]}|zdzfVdS)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%)))OO  :::: :::::: :::: :::::: 1q5 ::::q::::::: Hr)graphsrc|dkrtjd|dkrtjdSttjd|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. r zmust satisfy n >= 1r )r NetworkXError empty_graphr path_graph)rs rrrGsY@ 1uu4555Avv~a   2=++QU333r)r ) __doc__networkxrnetworkx.utilsr__all__ _dispatchablerrrrrr1s ...... + ,Z  \""%%%5 5 5 &%#"! 5 pT222&4&432&4&4&4r