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- """
- Ramsey numbers.
- """
- import networkx as nx
- from networkx.utils import not_implemented_for
- from ...utils import arbitrary_element
- __all__ = ["ramsey_R2"]
- @not_implemented_for("directed")
- @not_implemented_for("multigraph")
- def ramsey_R2(G):
- r"""Compute the largest clique and largest independent set in `G`.
- This can be used to estimate bounds for the 2-color
- Ramsey number `R(2;s,t)` for `G`.
- This is a recursive implementation which could run into trouble
- for large recursions. Note that self-loop edges are ignored.
- Parameters
- ----------
- G : NetworkX graph
- Undirected graph
- Returns
- -------
- max_pair : (set, set) tuple
- Maximum clique, Maximum independent set.
- Raises
- ------
- NetworkXNotImplemented
- If the graph is directed or is a multigraph.
- """
- if not G:
- return set(), set()
- node = arbitrary_element(G)
- nbrs = (nbr for nbr in nx.all_neighbors(G, node) if nbr != node)
- nnbrs = nx.non_neighbors(G, node)
- c_1, i_1 = ramsey_R2(G.subgraph(nbrs).copy())
- c_2, i_2 = ramsey_R2(G.subgraph(nnbrs).copy())
- c_1.add(node)
- i_2.add(node)
- # Choose the larger of the two cliques and the larger of the two
- # independent sets, according to cardinality.
- return max(c_1, c_2, key=len), max(i_1, i_2, key=len)
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