U 9v f™ ã@sFdZddlZddlmZddgZedƒedƒd d d„ƒƒZd d„ZdS) zTFunctions related to the Mycielski Operation and the Mycielskian family of graphs. éN)Únot_implemented_forÚ mycielskianÚmycielski_graphZdirectedZ multigraphécsžt |¡}t|ƒD]†}| ¡‰| tˆdˆƒ¡t| ¡ƒ}| ‡fdd„|Dƒ¡| ‡fdd„|Dƒ¡| dˆ¡| ‡fdd„tˆƒDƒ¡q|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. éc3s|]\}}||ˆfVqdS©N©©Ú.0ÚuÚv©ÚnrúA/tmp/pip-unpacked-wheel-_lngutwb/networkx/generators/mycielski.pyÚ >szmycielskian..c3s|]\}}|ˆ|fVqdSrrr r rrr?sc3s|]}|ˆdˆfVqdS)rNr)r r r rrrAs) ÚnxZconvert_node_labels_to_integersÚrangeZnumber_of_nodesZadd_nodes_fromÚlistÚedgesZadd_edges_fromÚadd_node)ÚGZ iterationsÚMÚiZ old_edgesrr rr s,   cCs<|dkrt d¡‚|dkr$t d¡Stt d¡|dƒSdS)a‡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. rzmust satisfy n >= 0rN)rZ NetworkXErrorZ empty_graphrZ path_graphr rrrrFs   )r)Ú__doc__ZnetworkxrZnetworkx.utilsrÚ__all__rrrrrrÚs 8