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"""Functions related to the Mycielski Operation and the Mycielskian family |
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of graphs. |
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""" |
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import networkx as nx |
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from networkx.utils import not_implemented_for |
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__all__ = ["mycielskian", "mycielski_graph"] |
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@not_implemented_for("directed") |
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@not_implemented_for("multigraph") |
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@nx._dispatchable(returns_graph=True) |
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def mycielskian(G, iterations=1): |
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r"""Returns the Mycielskian of a simple, undirected graph G |
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The Mycielskian of graph preserves a graph's triangle free |
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property while increasing the chromatic number by 1. |
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The Mycielski Operation on a graph, :math:`G=(V, E)`, constructs a new |
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graph with :math:`2|V| + 1` nodes and :math:`3|E| + |V|` edges. |
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The construction is as follows: |
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Let :math:`V = {0, ..., n-1}`. Construct another vertex set |
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:math:`U = {n, ..., 2n}` and a vertex, `w`. |
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Construct a new graph, `M`, with vertices :math:`U \bigcup V \bigcup w`. |
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For edges, :math:`(u, v) \in E` add edges :math:`(u, v), (u, v + n)`, and |
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:math:`(u + n, v)` to M. Finally, for all vertices :math:`u \in U`, add |
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edge :math:`(u, w)` to M. |
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The Mycielski Operation can be done multiple times by repeating the above |
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process iteratively. |
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More information can be found at https://en.wikipedia.org/wiki/Mycielskian |
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Parameters |
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---------- |
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G : graph |
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A simple, undirected NetworkX graph |
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iterations : int |
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The number of iterations of the Mycielski operation to |
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perform on G. Defaults to 1. Must be a non-negative integer. |
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Returns |
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------- |
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M : graph |
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The Mycielskian of G after the specified number of iterations. |
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Notes |
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----- |
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Graph, node, and edge data are not necessarily propagated to the new graph. |
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""" |
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M = nx.convert_node_labels_to_integers(G) |
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for i in range(iterations): |
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n = M.number_of_nodes() |
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M.add_nodes_from(range(n, 2 * n)) |
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old_edges = list(M.edges()) |
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M.add_edges_from((u, v + n) for u, v in old_edges) |
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M.add_edges_from((u + n, v) for u, v in old_edges) |
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M.add_node(2 * n) |
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M.add_edges_from((u + n, 2 * n) for u in range(n)) |
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return M |
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@nx._dispatchable(graphs=None, returns_graph=True) |
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def mycielski_graph(n): |
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"""Generator for the n_th Mycielski Graph. |
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The Mycielski family of graphs is an infinite set of graphs. |
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:math:`M_1` is the singleton graph, :math:`M_2` is two vertices with an |
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edge, and, for :math:`i > 2`, :math:`M_i` is the Mycielskian of |
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:math:`M_{i-1}`. |
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More information can be found at |
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http://mathworld.wolfram.com/MycielskiGraph.html |
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Parameters |
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---------- |
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n : int |
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The desired Mycielski Graph. |
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Returns |
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------- |
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M : graph |
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The n_th Mycielski Graph |
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Notes |
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----- |
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The first graph in the Mycielski sequence is the singleton graph. |
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The Mycielskian of this graph is not the :math:`P_2` graph, but rather the |
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:math:`P_2` graph with an extra, isolated vertex. The second Mycielski |
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graph is the :math:`P_2` graph, so the first two are hard coded. |
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The remaining graphs are generated using the Mycielski operation. |
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""" |
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if n < 1: |
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raise nx.NetworkXError("must satisfy n >= 1") |
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if n == 1: |
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return nx.empty_graph(1) |
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else: |
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return mycielskian(nx.path_graph(2), n - 2) |
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