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| """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"] | |
| 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. | |
| """ | |
| 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 | |
| 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 >= 1") | |
| if n == 1: | |
| return nx.empty_graph(1) | |
| else: | |
| return mycielskian(nx.path_graph(2), n - 2) | |